਍㄀⸀ ⠀甀⬀瘀⤀⬀眀 㴀 甀⬀⠀瘀⬀眀⤀㰀戀爀㸀ഀഀ 2. 0+u = u+0 = u  (where "0" is the nulvector, like for example (0,0) or (0,0,0) etc...
਍㌀⸀ 甀⬀⠀ⴀ甀⤀ 㴀　㰀戀爀㸀ഀഀ 4. u+v = v+u
਍㔀⸀ ☀⌀㤀㔀㔀㬀 ⠀甀⬀瘀⤀ 㴀 ☀⌀㤀㔀㔀㬀 甀 ⬀ ☀⌀㤀㔀㔀㬀 瘀㰀戀爀㸀ഀഀ 6. (λ + μ) v = λ v + μ v
਍㜀⸀ ⠀☀⌀㤀㔀㔀㬀 ☀⌀㤀㔀㘀㬀⤀ 瘀 㴀 ☀⌀㤀㔀㘀㬀 ⠀☀⌀㤀㔀㔀㬀 瘀⤀㰀戀爀㸀ഀഀ 8. 1 v = v  (where "1" is simply the scalar "1")
਍㰀⼀栀㌀㸀ഀഀ ਍吀栀攀猀攀 㠀 瀀爀漀瀀攀爀琀椀攀猀 漀昀 ∀嘀∀ 眀椀氀氀 渀漀琀 爀攀愀氀氀礀 愀洀愀稀攀 礀漀甀⸀ 䘀漀爀 洀漀猀琀 瀀爀漀瀀攀爀琀椀攀猀Ⰰ 眀攀 栀愀瘀攀 猀攀攀渀 攀砀愀洀瀀氀攀猀 椀渀 渀漀琀攀 ㄀㈀⸀㰀戀爀㸀ഀഀ Let me recapitulate one. Say, number 4.
਍匀甀瀀瀀漀猀攀 眀攀 挀漀渀猀椀搀攀爀 琀栀攀 瘀攀挀琀漀爀猀瀀愀挀攀 刀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀⸀ 匀甀瀀瀀漀猀攀 眀攀 栀愀瘀攀 琀栀攀 瘀攀挀琀漀爀猀 䄀 愀渀搀 䈀Ⰰ 氀椀欀攀㨀㰀戀爀㸀ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ
਍㰀吀䐀㸀䄀 㴀㰀戀爀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a₁ ┐
਍☀⌀㤀㐀㜀㐀㬀 愀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ a₃ ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀刀㸀ഀഀ ਍ഀഀ ਍ഀഀ ਍㰀吀刀㸀㰀戀爀㸀ഀഀ ਍ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀 ഀഀ
਍吀栀攀渀 䄀 ⬀ 䈀 㴀 䈀 ⬀ 䄀Ⰰ 眀栀椀挀栀 挀愀渀 戀攀 洀愀搀攀 瀀氀愀甀猀椀戀氀攀 ⠀漀爀 瀀爀漀瘀攀渀⤀ 戀礀㨀㰀戀爀㸀ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ
਍㰀吀䐀㸀䄀 ⬀ 䈀 㴀㰀戀爀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a₁ ┐
਍☀⌀㤀㐀㜀㐀㬀 愀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ a₃ ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀㴀㰀戀爀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a₁ + b₁ ┐
਍☀⌀㤀㐀㜀㐀㬀 愀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⬀ 戀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀  ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ a₃ + b₃ ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀㴀㰀戀爀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ b₁ + a₁ ┐
਍☀⌀㤀㐀㜀㐀㬀 戀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⬀ 愀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀  ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ b₃ + a₃ ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀㴀㰀戀爀㸀ഀഀ ਍㰀吀䐀㸀䈀 ⬀ 䄀㰀戀爀㸀ഀഀ ਍ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀 ഀഀ
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀甀攀∀㸀ഀഀ

1.2. What is a "Matrix"?:

਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍䄀 洀愀琀爀椀砀 椀猀 愀渀 ⠀爀攀挀琀愀渀最甀氀愀爀⤀ ∀愀爀爀愀礀∀ 漀昀 渀甀洀戀攀爀猀⸀ 䤀琀 栀愀猀 ∀洀∀ 爀漀眀猀Ⰰ 愀渀搀 ∀渀∀ 挀漀氀甀洀渀猀⸀㰀戀爀㸀ഀഀ In many applications, we have m=n, so the number of rows is equal to the number of columns. In that case, it's a "square" matrix.
਍㰀戀爀㸀ഀഀ Here is an example:
਍㰀戀爀㸀ഀഀ Figure 1. Example MxN matrix ("m" rows, and "n" columns)
਍㰀戀爀㸀ഀഀ ਍㰀戀爀㸀ഀഀ
਍䐀漀 渀漀琀 昀漀爀最攀琀 琀栀愀琀 琀栀攀 ∀攀氀攀洀攀渀琀猀∀ 氀椀欀攀 愀㰀猀甀戀㸀㄀㈀㰀⼀猀甀戀㸀Ⰰ 椀渀 琀栀攀 洀愀琀爀椀砀Ⰰ 愀爀攀 樀甀猀琀 渀甀洀戀攀爀猀Ⰰ 氀椀欀攀 昀漀爀 攀砀愀洀瀀氀攀 ㄀Ⰰ ⴀ㔀Ⰰ ㈀㜀⸀㌀㐀㐀Ⰰ ☀⌀㤀㘀　㬀 攀琀挀⸀⸀㰀戀爀㸀ഀഀ
਍伀昀挀漀甀爀猀攀Ⰰ 愀 洀愀琀爀椀砀 椀猀 渀漀琀 ∀樀甀猀琀∀ 愀渀 愀爀爀愀礀 漀昀 渀甀洀戀攀爀猀⸀ 䤀琀 洀甀猀琀 栀愀瘀攀 愀 ∀爀攀愀氀∀ 洀攀愀渀椀渀最 琀漀 椀琀⸀㰀戀爀㸀ഀഀ I am going to make it plausible for you, that a Matrix can be interpreted as:
਍㰀戀爀㸀ഀഀ Listing 1:
਍㰀戀爀㸀ഀഀ 1. An object that determines and describes all the coefficients of a set of linear equations.
਍㈀⸀ 䄀 ∀洀愀瀀瀀椀渀最∀ ⠀漀爀 伀瀀攀爀愀琀漀爀Ⰰ 漀爀 䰀椀渀攀愀爀 吀爀愀渀猀昀漀爀洀愀琀椀漀渀⤀ 漀渀 瘀攀挀琀漀爀猀⸀㰀戀爀㸀ഀഀ 3. That it can represents a tensor (in some fields of math or physics).
਍㰀戀爀㸀ഀഀ That a matrix can be identified as a linear "mapping" (sort of function on vectors), is probably
਍琀栀攀 洀漀猀琀 挀漀洀洀漀渀 椀搀攀愀 漀渀 洀愀琀爀椀挀攀猀⸀㰀戀爀㸀ഀഀ
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀㸀ഀഀ Note:
਍䤀渀 瀀爀漀昀攀猀猀椀漀渀愀氀 氀椀琀攀爀愀琀甀爀攀Ⰰ 猀挀椀攀渀琀椀猀琀猀 漀昀琀攀渀 甀猀攀 愀 猀漀爀琀 漀昀 ∀猀栀漀爀琀挀甀琀∀ 渀漀琀愀琀椀漀渀 昀漀爀 愀 洀愀琀爀椀砀⸀㰀戀爀㸀ഀഀ Instead of the large rectangular array, they might simply write the matrix as:
਍ഀഀ

  a_ij

਍ഀഀ where it is "implicitly" assumed (or clear) that we know that i "runs" from "1 to m", and j "runs" from "1 to n".
਍䤀 眀椀氀氀 渀漀琀 甀猀攀 椀琀 洀甀挀栀Ⰰ 戀甀琀 愀猀 礀漀甀 洀椀最栀琀 猀攀攀Ⰰ 椀琀✀猀 愀 最爀攀愀琀 眀愀礀 渀漀琀 琀漀 栀愀瘀攀 琀栀攀 琀爀漀甀戀氀攀 漀昀 眀爀椀琀椀渀最 搀漀眀渀 猀甀挀栀 愀 ∀氀愀爀最攀 漀戀樀攀挀琀∀㰀戀爀㸀ഀഀ as a N x M matrix, like in figure 1. Saves a lot of time and effort.
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍ഀഀ
਍䈀攀猀椀搀攀猀 琀栀愀琀 眀攀 愀爀攀 最漀椀渀最 琀漀 甀渀搀攀爀猀琀愀渀搀 栀漀眀 琀漀 椀渀琀攀爀瀀爀攀琀 洀愀琀爀椀挀攀猀Ⰰ 眀攀 漀昀挀漀甀爀猀攀 眀椀氀氀 搀漀 猀漀洀攀 最爀攀愀琀 攀砀挀攀爀挀椀猀攀猀 琀漀漀⸀㰀戀爀㸀ഀഀ
਍ഀഀ ਍㰀栀㈀㸀㄀⸀㌀ 吀栀攀 洀愀琀爀椀砀 愀猀 愀 ∀挀漀攀昀昀椀挀椀攀渀琀 洀愀琀爀椀砀∀ 漀昀 愀 猀攀琀 氀椀渀攀愀椀爀 攀焀甀愀琀椀漀渀猀㰀⼀栀㈀㸀ഀഀ ਍ഀഀ ਍㰀栀㌀㸀䔀砀愀洀瀀氀攀㨀 挀漀漀爀搀椀渀愀琀攀 琀爀愀渀猀昀漀爀洀愀琀椀漀渀⸀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ I'am going to use a "famous" problem here: a coordinate transformation. The discussion will show that a matrix is indeed
਍愀 ∀洀愀瀀瀀椀渀最∀Ⰰ 愀猀 眀攀氀氀 愀猀 琀栀愀琀 礀漀甀 挀愀渀 椀渀琀攀爀瀀爀攀琀 椀琀 愀猀 愀渀 漀戀樀攀挀琀 琀栀愀琀 搀攀琀攀爀洀椀渀攀猀 愀氀氀 琀栀攀 挀漀攀昀昀椀挀椀攀渀琀猀 漀昀 愀 猀攀琀 漀昀 氀椀渀攀愀爀 攀焀甀愀琀椀漀渀猀⸀㰀戀爀㸀ഀഀ
਍匀甀瀀瀀漀猀攀 眀攀 栀愀瘀攀 愀 瘀攀挀琀漀爀猀瀀愀挀攀 ∀嘀∀ 漀昀 搀椀洀攀渀猀椀漀渀 ∀渀∀⸀ 吀栀攀渀 眀攀 渀攀攀搀 ∀渀∀ 椀渀搀攀瀀攀渀搀攀渀搀 戀愀猀椀猀瘀攀挀琀漀爀猀 ⠀漀爀 甀渀椀琀 瘀攀挀琀漀爀猀⤀Ⰰ㰀戀爀㸀ഀഀ in order to be able to describe any other vector.
਍䤀琀✀猀 焀甀椀琀攀 琀栀攀 猀愀洀攀 愀猀 眀攀 愀氀爀攀愀搀礀 猀愀眀 椀渀 渀漀琀攀 ㄀㈀ ⠀椀昀 渀攀攀搀攀搀Ⰰ 爀攀瘀椀猀椀琀 渀漀琀攀 ㄀㈀ 愀最愀椀渀⤀⸀㰀戀爀㸀ഀഀ
਍䈀甀琀 愀挀琀甀愀氀氀礀 眀攀 挀愀渀 栀愀瘀攀 洀甀氀琀椀瀀氀攀 ⠀甀渀氀椀洀椀琀攀搀⤀ 漀昀 猀甀挀栀 猀攀琀猀 漀昀 戀愀猀椀猀瘀攀挀琀漀爀猀⸀ 匀甀瀀瀀漀猀攀 琀栀愀琀 眀攀 氀漀漀欀 愀琀 琀栀攀 昀愀洀椀氀椀愀爀 刀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀 猀瀀愀挀攀⸀㰀戀爀㸀ഀഀ Then it's quite common to take (1,0,0), (0,1,0), and (0,0,1) as such basisvectors.
਍䈀甀琀 琀栀椀猀 椀猀 渀漀琀 琀栀攀 漀渀氀礀 瀀漀猀猀椀戀氀攀 猀攀琀⸀ 䨀甀猀琀 椀洀愀最椀渀最 琀栀愀琀 礀漀甀 ∀爀漀琀愀琀攀∀ 愀氀氀 愀砀攀猀 ⠀砀ⴀⰀ 礀Ⰰ 愀渀搀 稀ⴀ愀砀椀猀⤀ 漀瘀攀爀 ☀⌀㤀㘀　㬀⼀㈀ 搀攀最爀攀攀猀㰀戀爀㸀ഀഀ in some direction. ਍吀栀攀渀Ⰰ 礀漀甀 挀愀渀 猀攀攀 琀栀愀琀 琀栀攀爀攀 椀猀 愀渀漀琀栀攀爀 猀攀琀 漀昀 甀渀椀琀 瘀攀挀琀漀爀猀Ⰰ 琀椀氀琀攀搀 戀礀 ☀⌀㤀㘀　㬀⼀㈀Ⰰ 挀漀洀瀀愀爀攀搀 琀漀 漀甀爀 昀椀爀猀琀 漀渀攀⸀㰀戀爀㸀ഀഀ
਍匀漀Ⰰ 猀甀瀀瀀漀猀攀 眀攀 栀愀瘀攀 琀栀攀 昀漀氀氀漀眀椀渀最 琀眀漀 猀攀琀猀 漀昀 甀渀椀琀 瘀攀挀琀漀爀猀 漀昀 嘀㨀㰀戀爀㸀ഀഀ
਍㰀䈀㸀䰀椀猀琀椀渀最 ㈀㨀㰀⼀䈀㸀㰀戀爀㸀ഀഀ ਍㰀栀㈀㸀☀⌀㤀㄀㐀㬀 㴀 笀瘀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀Ⰰ 瘀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀Ⰰ ⸀⸀ Ⰰ瘀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀紀㰀戀爀㸀ഀഀ
਍☀⌀㤀㄀㐀㬀✀ 㴀 笀眀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀Ⰰ 眀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀Ⰰ ⸀⸀ Ⰰ眀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀紀㰀⼀栀㈀㸀ഀഀ ਍吀栀攀渀 㰀䈀㸀攀愀挀栀㰀⼀䈀㸀 眀㰀猀甀戀㸀椀㰀⼀猀甀戀㸀 挀愀渀 戀攀 瘀椀攀眀攀搀 愀猀 愀渀 㰀䈀㸀漀爀搀椀渀愀爀礀 瘀攀挀琀漀爀㰀⼀䈀㸀 椀渀 琀栀攀 猀攀琀 ☀⌀㤀㄀㐀㬀Ⰰ 愀渀搀 琀栀甀猀 挀愀渀 戀攀 攀砀瀀爀攀猀猀攀搀㰀戀爀㸀ഀഀ as a linear combination of the v₁, v₂, .. ,v_n basis vectors.
਍䤀渀 昀愀挀琀Ⰰ 眀攀 琀栀甀猀 漀戀琀愀椀渀 愀 ∀猀攀琀 漀昀 氀椀渀攀愀爀 攀焀甀愀琀椀漀渀猀∀⸀㰀戀爀㸀ഀഀ
਍㰀䈀㸀䰀椀猀琀椀渀最 ㌀㨀㰀⼀䈀㸀㰀戀爀㸀ഀഀ ਍㰀栀㈀㸀眀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀 㴀 愀㰀猀甀戀㸀㄀㄀㰀⼀猀甀戀㸀瘀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀 ⬀ 愀㰀猀甀戀㸀㄀㈀㰀⼀猀甀戀㸀瘀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⬀ ⸀⸀ ⬀ 愀㰀猀甀戀㸀㄀渀㰀⼀猀甀戀㸀瘀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀㰀戀爀㸀ഀഀ ..
਍⸀⸀㰀戀爀㸀ഀഀ w_n = a_n1v₁ + a_n2v₂ + .. + a_nnv_n ਍ഀഀ All those coefficients a_ij form a square nxn matrix.
਍㰀戀爀㸀ഀഀ Figure 2. the NxN matrix that belongs to listing 3.
਍㰀戀爀㸀ഀഀ ਍㰀戀爀㸀ഀഀ
਍ഀഀ ਍ഀഀ ਍匀漀Ⰰ 琀栀攀 挀漀漀爀搀椀渀愀琀攀 㰀䈀㸀琀爀愀渀猀昀漀爀洀愀琀椀漀渀㰀⼀䈀㸀 ⠀眀栀椀挀栀 椀猀 愀 洀愀瀀瀀椀渀最 琀漀漀⤀ 漀昀 漀甀爀 攀砀愀洀瀀氀攀 愀戀漀瘀攀Ⰰ 氀攀愀搀猀 琀漀 愀 猀攀琀 漀昀 氀椀渀攀愀爀 攀焀甀愀琀椀漀渀Ⰰ㰀戀爀㸀ഀഀ and we can have all coefficients of that set, correspond to a Matrix.
਍㰀戀爀㸀ഀഀ ਍ഀഀ
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀甀攀∀㸀ഀഀ

1.4 The matrix as a "Mapping" or Linear Transformation

਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍吀栀椀猀 挀栀愀瀀琀攀爀 椀猀 漀渀礀 愀渀 椀渀琀爀漀搀甀挀琀椀漀渀⸀ 䤀 眀愀渀琀 琀漀 攀猀琀愀戀氀椀猀栀 愀 挀攀爀琀愀椀渀 ∀氀漀漀欀 愀渀搀 昀攀攀氀∀ 愀琀 愀渀 攀愀爀氀礀 洀漀洀攀渀琀⸀㰀戀爀㸀ഀഀ A more thourough discussion follows at a later moment.
਍㰀戀爀㸀ഀഀ ਍圀攀 挀愀渀 洀甀氀琀椀瀀氀礀 愀渀 渀 砀 渀 洀愀琀爀椀砀 眀椀琀栀 愀渀 ㄀砀渀 䌀漀氀甀洀渀 嘀攀挀琀漀爀⸀㰀戀爀㸀ഀഀ It will turn out, that in many cases we can identify that multiplication with a "mapping", meaning
਍琀栀愀琀 愀 瘀攀挀琀漀爀 眀椀氀氀 戀攀 洀愀瀀瀀攀搀 琀漀 愀渀漀琀栀攀爀 瘀攀挀琀漀爀⸀㰀戀爀㸀ഀഀ
਍䤀 洀甀猀琀 猀愀礀 琀栀愀琀 琀栀攀 猀琀愀琀攀洀攀渀琀 㰀䤀㸀∀圀攀 挀愀渀 洀甀氀琀椀瀀氀礀 愀渀 渀 砀 渀 洀愀琀爀椀砀 眀椀琀栀 愀渀 ㄀砀渀 䌀漀氀甀洀渀 嘀攀挀琀漀爀∀㰀⼀䤀㸀 椀猀 渀漀琀㰀戀爀㸀ഀഀ a universal enough. As we will see, we can multiply a "MxN" matrix with a "NxS" matrix, leading to a "MxS" matrix.
਍一漀琀攀 栀漀眀 最攀渀攀爀愀氀 猀甀挀栀 猀琀愀琀攀洀攀渀琀 椀猀⸀ 吀栀攀 爀攀猀甀氀琀猀 挀漀甀氀搀 琀栀甀猀 愀氀猀漀 爀攀琀甀爀渀 愀 挀漀氀甀洀渀 漀爀 爀漀眀 瘀攀挀琀漀爀Ⰰ㰀戀爀㸀ഀഀ which in many cases can be identified as a mapping too.
਍㰀戀爀㸀ഀഀ We must make that "plausible" ofcourse, meaning that the statements really make sense.
਍㰀戀爀㸀ഀഀ However, for the moment, please take notice of the following two statements:
਍ഀഀ ਍㰀栀㌀㸀匀琀愀琀攀洀攀渀琀 ㄀㨀 最攀渀攀爀愀氀 洀甀氀琀椀瀀氀椀挀愀琀椀漀渀㰀⼀栀㌀㸀ഀഀ M x N . N x S => leads to a M x S matrix ਍ഀഀ It's quite easy to remember, since the result matrix uses the "outer" indices (m and s) only.
਍䤀琀 眀椀氀氀 戀攀 瀀爀漀瘀攀渀 愀琀 愀 氀愀琀攀爀 洀漀洀攀渀琀⸀㰀戀爀㸀ഀഀ
਍ഀഀ ਍㰀栀㌀㸀匀琀愀琀攀洀攀渀琀 ㈀㨀 琀眀漀 猀瀀攀挀椀愀氀 挀愀猀攀猀⸀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ

2.1 A 3x3 matrix and a column vector (1x3):

਍ഀഀ ਍㰀戀爀㸀ഀഀ Let's explain how it's done. Each element of each row of the matrix, starting with the first row, is multiplied
਍眀椀琀栀 琀栀攀 攀氀攀洀攀渀琀猀 漀昀 琀栀攀 挀漀氀甀洀渀 瘀攀挀琀漀爀⸀ 吀栀椀猀 猀栀漀甀氀搀 戀攀 焀甀椀琀攀 攀愀猀礀 琀漀 爀攀洀攀戀攀爀 琀漀漀⸀㰀戀爀㸀ഀഀ Note that the operation leads to another column vector.
਍圀攀 挀愀渀 椀搀攀渀琀椀昀礀 猀甀挀栀 洀愀琀爀椀砀 愀猀 愀 䰀椀渀攀愀爀 吀爀愀渀猀昀漀爀洀愀琀椀漀渀 漀渀 漀戀樀攀挀琀猀 椀渀 刀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀Ⰰ 猀漀 椀渀 最攀渀攀爀愀氀 眀攀 猀瀀攀愀欀 漀昀 愀 刀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀 ⴀ㸀 刀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀 洀愀瀀瀀椀渀最⸀㰀戀爀㸀ഀഀ
਍㰀栀㌀㸀㈀⸀㈀ 䄀 ㈀砀㈀ 洀愀琀爀椀砀 愀渀搀 愀 挀漀氀甀洀渀 瘀攀挀琀漀爀 ⠀㄀砀㈀⤀㨀㰀⼀栀㌀㸀ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ ax+by ┐
਍☀⌀㤀㐀㤀㈀㬀 挀砀⬀搀礀 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍䤀琀✀猀 焀甀椀琀攀 琀栀攀 猀愀洀攀 愀猀 ㈀⸀㄀Ⰰ 栀漀眀攀瘀攀爀 栀攀爀攀 眀攀 挀愀渀 猀攀攀 愀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 ⴀ㸀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 洀愀瀀瀀椀渀最⸀㰀戀爀㸀ഀഀ Note that here too, the result of the mapping is a vector again.
਍㰀戀爀㸀ഀഀ Examples:
਍㰀戀爀㸀ഀഀ Let's try a few examples of the procedure of 2.2:
਍㰀戀爀㸀ഀഀ Example 1:
਍㰀戀爀㸀ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 3*1+0*2 ┐
਍☀⌀㤀㐀㤀㈀㬀 　⨀㈀⬀㌀⨀㈀ ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍匀漀Ⰰ 椀渀 琀栀椀猀 攀砀愀洀瀀氀攀Ⰰ 琀栀攀 瘀攀挀琀漀爀 ⠀㄀⸀㈀⤀ 椀猀 洀愀瀀瀀攀搀 琀漀 ⠀㌀Ⰰ㘀⤀⸀㰀戀爀㸀ഀഀ Actually, this will happen with any vector in R². So, this mapping is a sort of "scaling" operator,
਍∀攀渀氀愀爀最椀渀最∀ 愀渀礀 瘀攀挀琀漀爀 戀礀 ∀㌀∀⸀㰀戀爀㸀ഀഀ
਍ഀഀ ਍ഀഀ ਍㰀䈀㸀䔀砀愀洀瀀氀攀 ㈀㨀㰀⼀䈀㸀㰀戀爀㸀ഀഀ
਍夀漀甀 搀漀 渀漀琀 渀攀攀搀 琀漀 ∀瘀攀爀椀昀礀∀ 漀爀 猀漀洀攀琀栀椀渀最Ⰰ 琀栀攀 昀漀氀氀漀眀椀渀最 洀愀瀀瀀椀渀最⸀㰀戀爀㸀ഀഀ Let's again stay in R². If we have any vector (x,y), then a clockwise rotation
਍漀昀 琀栀攀 瘀攀挀琀漀爀 漀瘀攀爀 愀渀 愀渀最氀攀 ☀⌀㤀㠀㄀㬀Ⰰ 挀愀渀 戀攀 攀砀瀀爀攀猀猀攀搀 戀礀㨀㰀戀爀㸀ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ
਍ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍匀漀Ⰰ 椀昀 椀渀搀攀攀搀 眀漀甀氀搀 栀愀瘀攀 愀 瀀愀爀琀椀挀甀氀愀爀 瘀攀挀琀漀爀Ⰰ 猀愀礀 ⠀㄀Ⰰ㈀⤀ 漀爀 愀渀礀 漀琀栀攀爀Ⰰ 愀渀搀 礀漀甀 欀渀漀眀 琀栀攀 愀渀最氀攀 ☀⌀㤀㠀㄀㬀Ⰰ㰀戀爀㸀ഀഀ then you can calculate sin(ϕ) and cos(ϕ). Then using the general procedure as shown in 2.2,
਍愀氀氀漀眀猀 礀漀甀 琀漀 挀愀氀挀甀氀愀琀攀 琀栀攀 爀攀猀甀氀琀 瘀攀挀琀漀爀⸀㰀戀爀㸀ഀഀ
਍刀攀洀攀洀戀攀爀Ⰰ 琀栀攀 洀愀椀渀 瀀甀爀瀀漀猀攀 漀昀 琀栀椀猀 挀栀愀瀀琀攀爀Ⰰ 椀猀 琀漀 最椀瘀攀 礀漀甀 愀 瀀爀攀琀琀礀 最漀漀搀 椀搀攀愀 漀渀 琀栀攀 猀攀瘀攀爀愀氀 椀渀琀攀爀瀀爀攀琀愀琀椀漀渀猀㰀戀爀㸀ഀഀ of matrices (as was shown in listing 1 above). And at this point, real "calculations" are not important.
਍㰀戀爀㸀ഀഀ ਍㰀栀㈀㸀㄀⸀㔀 䴀愀琀爀椀挀攀猀 愀渀搀 吀攀渀猀漀爀猀⸀㰀⼀栀㈀㸀ഀഀ ਍ഀഀ Maybe it's not appropriate to say something on tensors at this stage, but I like to try
਍琀漀 瀀爀漀瘀椀搀攀 愀 最漀漀搀 漀瘀攀爀瘀椀攀眀 漀渀 洀愀琀琀攀爀猀⸀㰀戀爀㸀ഀഀ
਍䄀猀 椀琀 椀猀 渀漀眀Ⰰ 眀攀 栀愀瘀攀 猀挀愀氀愀爀猀Ⰰ 瘀攀挀琀漀爀猀Ⰰ 愀渀搀 洀愀琀爀椀挀攀猀⸀㰀戀爀㸀ഀഀ
਍ⴀ猀挀愀氀愀爀㨀 樀甀猀琀 愀 渀甀洀戀攀爀Ⰰ 氀椀欀攀 琀栀攀 琀攀洀瀀攀爀愀琀甀爀攀 愀琀 瀀漀椀渀琀猀 椀渀 猀瀀愀挀攀⸀㰀戀爀㸀ഀഀ There is no "direction" associated with scalars.
਍㰀戀爀㸀ഀഀ -vector: an object with magnitude (length) and direction.
਍䴀愀渀礀 瀀栀礀猀椀挀愀氀 瀀栀攀渀漀洀攀渀愀 挀愀渀 戀攀 攀砀瀀爀攀猀猀攀搀 戀礀 瘀攀挀琀漀爀猀Ⰰ 氀椀欀攀 愀渀 䔀氀攀挀琀爀椀挀 昀椀攀氀搀Ⰰ㰀戀爀㸀ഀഀ which has a certain magnitude at points, but a "direction" as well (e.g. a positively charge partice moves along the field).
਍㰀戀爀㸀ഀഀ -Matrix: It's probably best to associate it with an "operator", or "linear mapping".
਍㰀戀爀㸀ഀഀ Well, it's math so take stuff seriously. However, not too "strickt"....
਍㰀戀爀㸀ഀഀ You can easily say that some scalar, is an operator as well. Just look at this:
਍㰀戀爀㸀ഀഀ

B = ਍㰀⼀吀䐀㸀ഀഀ	਍☀⌀㤀㐀㠀㐀㬀 戀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ b2 │ ਍☀⌀㤀㐀㤀㈀㬀 戀㰀猀甀戀㸀㌀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㤀㘀㬀ഀഀ
+ ਍㰀⼀吀䐀㸀ഀഀ	਍☀⌀㤀㐀㠀㐀㬀 戀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ b2 │ ਍☀⌀㤀㐀㤀㈀㬀 戀㰀猀甀戀㸀㌀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㤀㘀㬀ഀഀ
਍☀⌀㤀㐀㠀㐀㬀 愀 戀 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ c d ┘ ਍㰀⼀吀䐀㸀ഀഀ	਍☀⌀㤀㐀㠀㐀㬀 砀 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ y ┘ ਍㰀⼀吀䐀㸀ഀഀ	਍㴀ഀഀ	਍☀⌀㤀㐀㠀㐀㬀 砀✀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ y' ┘ ਍㰀⼀吀䐀㸀ഀഀ
਍☀⌀㤀㐀㠀㐀㬀 ㌀ 　 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 0 3 ┘ ਍㰀⼀吀䐀㸀ഀഀ	਍☀⌀㤀㐀㠀㐀㬀 ㄀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 2 ┘ ਍㰀⼀吀䐀㸀ഀഀ	਍㴀ഀഀ	਍☀⌀㤀㐀㠀㐀㬀 ㌀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 6 ┘ ਍㰀⼀吀䐀㸀ഀഀ
਍☀⌀㤀㐀㠀㐀㬀 挀漀猀⠀☀⌀㤀㠀㄀㬀⤀⸀ 猀椀渀⠀☀⌀㤀㠀㄀㬀⤀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ -sin(ϕ) cos(ϕ) ┘ ਍㰀⼀吀䐀㸀ഀഀ	਍☀⌀㤀㐀㠀㐀㬀 砀 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ y ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ 5 * ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 15 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀　 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀⼀吀刀㸀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 ㌀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 2 ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍㴀ഀഀ

਍㰀戀爀㸀ഀഀ Here the scalar "5" looks like a (linear) operator... and here it really is.
਍㰀戀爀㸀ഀഀ Now, about tensors. Tensors have a "rank". A tensor of rank 2, can be identified as a matrix.
਍㰀戀爀㸀ഀഀ You know that a matrix is characterized by it's elements. We generally have a mxn Matrix,
਍眀椀琀栀 洀 爀漀眀猀 愀渀搀 渀 挀漀氀甀洀渀猀⸀ 圀攀 渀攀攀搀 琀眀漀 椀渀搀椀挀攀猀Ⰰ 琀漀 爀甀渀 漀瘀攀爀 愀氀氀 洀 愀渀搀 渀⸀㰀戀爀㸀ഀഀ
਍匀漀Ⰰ 眀椀琀栀 愀㰀猀甀戀㸀椀樀㰀⼀猀甀戀㸀㴀愀㰀猀甀戀㸀㈀㌀㰀⼀猀甀戀㸀 眀攀 椀渀搀攀渀琀椀昀礀 琀栀攀 攀氀攀洀攀渀琀 椀渀 琀栀攀 猀攀挀漀渀搀 爀漀眀Ⰰ 琀栀攀 琀栀椀爀搀 挀漀氀甀洀渀⸀㰀戀爀㸀ഀഀ
਍䈀甀琀 琀攀渀猀漀爀猀 攀砀椀猀琀 眀椀琀栀 攀瘀攀渀 愀 栀椀最栀攀爀 爀愀渀欀Ⰰ 昀漀爀 攀砀愀洀瀀氀攀 爀愀渀欀 ㌀⸀㰀戀爀㸀ഀഀ
਍匀漀洀攀 猀琀爀甀挀琀甀爀攀猀 椀渀 瀀栀礀猀椀挀猀Ⰰ 挀愀渀 漀渀氀礀 愀搀攀焀甀愀琀攀氀礀 戀攀 搀攀猀挀爀椀戀攀搀 戀礀 愀 爀愀渀欀 ㌀ 琀攀渀猀漀爀⸀㰀戀爀㸀ഀഀ For example, SpaceTime. Einstein described gravity, as the result of curved SpaceTime.
਍䠀攀 攀砀琀攀渀猀椀瘀攀氀礀 甀猀攀搀 栀椀最栀攀爀 爀愀渀欀 琀攀渀猀漀爀猀 椀渀 琀栀攀 琀栀攀漀爀椀攀猀⸀㰀戀爀㸀ഀഀ
਍䈀甀琀 眀攀 挀愀渀 猀琀愀礀 挀氀漀猀攀爀 琀漀 栀漀洀攀 愀猀 眀攀氀氀⸀ 匀甀瀀瀀漀猀攀 礀漀甀 栀愀瘀攀 猀漀洀攀 洀愀琀攀爀椀愀氀Ⰰ 氀椀欀攀 愀 戀氀漀挀欀 漀昀 洀攀琀愀氀⸀㰀戀爀㸀ഀഀ I you apply a torque at both ends, then the inner "twist" of directions that the material feels,
਍椀猀 猀漀 挀漀洀瀀氀攀砀 琀栀愀琀 礀漀甀 渀攀攀搀 愀 猀漀爀琀 漀昀 愀㰀猀甀戀㸀椀樀欀㰀⼀猀甀戀㸀 洀愀琀栀攀洀愀琀椀挀愀氀 漀戀樀攀挀琀Ⰰ 琀漀 搀攀猀挀爀椀戀攀 椀琀 洀愀琀栀攀洀愀琀椀挀愀氀氀礀⸀㰀戀爀㸀ഀഀ
਍䈀甀琀 眀愀椀琀Ⰰ 眀攀 挀愀渀 漀渀氀礀 栀愀瘀攀 洀砀渀 洀愀琀爀椀挀攀猀Ⰰ 搀攀猀挀爀椀戀攀搀 戀礀 攀氀攀洀攀渀琀猀 愀㰀猀甀戀㸀椀樀㰀⼀猀甀戀㸀⸀ 匀漀 挀氀攀愀爀氀礀Ⰰ 愀渀 愀㰀猀甀戀㸀椀樀欀㰀⼀猀甀戀㸀 琀攀渀猀漀爀㰀戀爀㸀ഀഀ is really beyond a normal mxn matrix.
਍㰀戀爀㸀ഀഀ For tensors (in physics), ijk indices can be used, but more often, physicists like indices as μ, ν η.
਍匀漀Ⰰ 椀渀 瀀爀漀昀攀猀猀椀漀渀愀氀 氀椀琀攀爀愀琀甀爀攀Ⰰ 礀漀甀 洀愀礀 猀攀攀 愀 琀栀椀爀搀 爀愀渀欀 琀攀渀猀漀爀 愀猀 吀㰀猀甀戀㸀☀⌀㤀㔀㄀㬀㰀⼀猀甀戀㸀㰀猀甀瀀㸀☀⌀㤀㔀㘀㬀 ☀⌀㤀㔀㜀㬀㰀⼀猀甀瀀㸀 漀爀 吀㰀猀甀瀀㸀☀⌀㤀㔀㘀㬀 ☀⌀㤀㔀㜀㬀 ☀⌀㤀㔀㄀㬀㰀⼀猀甀瀀㸀⸀㰀戀爀㸀ഀഀ
਍䄀 琀栀椀爀搀 爀愀渀欀 琀攀渀猀漀爀 洀愀礀 戀攀 瘀椀猀甀愀氀椀稀攀搀 愀猀 愀 ∀猀琀愀挀欀∀ 漀昀 愀㰀猀甀戀㸀椀樀㰀⼀猀甀戀㸀 洀愀琀爀椀挀攀猀Ⰰ 漀爀 戀攀琀琀攀爀Ⰰ 愀猀 愀 挀甀戀攀⸀㰀戀爀㸀ഀഀ For higher rank tensors, we cannot visualize it anymore. We can only do math with them.
਍㰀戀爀㸀ഀഀ The figure below, might give an impresson on a (second rank) matrix, and a third rank tensor.
਍㰀戀爀㸀ഀഀ Keep in mind, the nxm does not determine the rank of a true matrix: it's rank is always 2 (or 1 in case
਍漀昀 愀 ㄀砀渀 漀爀 渀砀㄀ 洀愀琀爀椀砀Ⰰ 眀栀攀爀攀 礀漀甀 漀渀氀礀 渀攀攀搀 漀渀攀 椀渀搀攀砀⤀⸀㰀戀爀㸀ഀഀ
਍匀漀Ⰰ 攀瘀攀渀 椀昀 礀漀甀 栀愀瘀攀 愀 㠀砀㄀　 洀愀琀爀椀砀Ⰰ 礀漀甀 猀琀椀氀氀 栀愀瘀攀 漀渀氀礀 ㈀ 椀渀搀椀挀攀猀Ⰰ ∀椀∀ 愀渀搀 ∀樀∀⸀ 䤀渀搀攀攀搀Ⰰ 礀漀甀 猀琀椀氀氀 漀渀氀礀 栀愀瘀攀 琀漀 搀攀愀氀㰀戀爀㸀ഀഀ with rows and columns. A matrix will always resemble a part of a spreadsheet if you like.
਍㰀戀爀㸀ഀഀ Only a rank 3 tensor, or higer rank tensor, needs more than 2 indices to describe the elements.
਍䄀 爀愀渀欀 ㌀ 琀攀渀猀漀爀Ⰰ 氀漀漀欀猀 氀椀欀攀 愀 ∀挀甀戀攀∀Ⰰ 漀爀 猀琀愀挀欀 漀昀 洀愀琀爀椀挀攀猀Ⰰ 眀栀攀爀攀 琀栀甀猀 漀渀攀 攀砀琀爀愀 椀渀搀攀砀 椀猀 爀攀焀甀椀爀攀搀⸀㰀戀爀㸀ഀഀ
਍㰀䤀㸀吀栀攀 愀戀漀瘀攀 琀攀砀琀Ⰰ 椀猀 欀攀瀀琀 爀愀琀栀攀爀 最攀渀攀爀愀氀Ⰰ 愀渀搀 椀渀搀攀攀搀Ⰰ 琀栀攀爀攀 愀爀攀 猀漀洀攀 猀瀀攀挀椀愀氀 攀砀挀攀瀀琀椀漀渀猀⸀㰀⼀䤀㸀㰀戀爀㸀ഀഀ
਍㰀䈀㸀䘀椀最甀爀攀 ㌀⸀ 䌀漀洀瀀愀爀椀渀最 愀 猀攀挀漀渀搀 爀愀渀欀 琀攀渀猀漀爀 ⠀漀爀 洀愀琀爀椀砀⤀Ⰰ 眀椀琀栀 愀 琀栀椀爀搀 爀愀渀欀 琀攀渀猀漀爀⸀㰀⼀䈀㸀㰀戀爀㸀ഀഀ
਍㰀椀洀最 猀爀挀㴀∀瘀瀀愀爀琀㈀开㐀⸀樀瀀最∀ 愀氀椀最渀㴀∀挀攀渀琀爀攀∀⼀㸀ഀഀ
਍㰀戀爀㸀ഀഀ ਍吀栀攀 渀攀砀琀 挀栀愀瀀琀攀爀 搀攀愀氀猀 眀椀琀栀 昀甀爀琀栀攀爀 瀀爀漀瀀攀爀琀椀攀猀 漀昀 洀愀琀爀椀挀攀猀Ⰰ 愀渀搀 洀愀琀爀椀砀 挀愀氀挀甀氀愀琀椀漀渀猀⸀㰀戀爀㸀ഀഀ
਍㰀戀爀㸀ഀഀ ਍㰀栀㄀㸀䌀栀愀瀀琀攀爀 ㈀⸀ 倀爀漀瀀攀爀琀椀攀猀 漀昀 洀愀琀爀椀挀攀猀Ⰰ 愀渀搀 挀愀氀挀甀氀愀琀椀渀最 眀椀琀栀 洀愀琀爀椀挀攀猀㨀㰀⼀栀㄀㸀ഀഀ ਍ഀഀ If we see a matrix operating on a vector, no matter what the dimension the Vectorspace is,
਍氀椀欀攀 昀漀爀 攀砀愀洀瀀氀攀㨀㰀戀爀㸀ഀഀ
਍ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a b ┐
਍☀⌀㤀㐀㤀㈀㬀 挀 搀 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ x ┐
਍☀⌀㤀㐀㤀㈀㬀 礀 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ ਍㰀戀爀㸀ഀഀ Then we will often use the following condensed notation:
਍ഀഀ

AX = X'

਍ഀഀ Where A is the matrix, X is the original vector, and X' is the (mapped) resultvector.
਍㰀戀爀㸀ഀഀ Next, we will see a listing of short subjects, all of which are important in "vector calculus" or "linear algebra".
਍㰀戀爀㸀ഀഀ ਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀甀攀∀㸀ഀഀ

1. Multiplication of a matrix with a scalar:

਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍匀甀瀀瀀漀猀攀 䄀 椀猀 愀 洀愀琀爀椀砀Ⰰ 愀渀搀 ☀⌀㤀㔀㔀㬀 椀猀 愀 猀挀愀氀愀爀 ⠀愀 猀挀愀氀愀爀 椀猀 樀甀猀琀 愀 渀甀洀戀攀爀⤀⸀㰀戀爀㸀ഀഀ
਍吀栀攀渀㨀 洀甀氀琀椀瀀氀礀椀渀最 琀栀攀 洀愀琀爀椀砀 眀椀琀栀 ☀⌀㤀㔀㔀㬀Ⰰ 洀攀愀渀猀 洀甀氀琀椀瀀氀礀椀渀最 攀愀挀栀 攀氀攀洀攀渀琀 愀㰀猀甀戀㸀椀樀㰀⼀猀甀戀㸀 眀椀琀栀 ☀⌀㤀㔀㔀㬀⸀㰀戀爀㸀ഀഀ So, for example for a 2x2 matrix:
਍ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 砀✀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ y' ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ A = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍ഀഀ then λ A would yield:
਍ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 愀 戀 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ c d ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ λ A = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ ਍ഀഀ ਍㰀栀㌀㸀㈀⸀ 䄀搀搀椀琀椀漀渀 漀昀 洀愀琀爀椀挀攀猀㨀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ This is only defined and meningful, if both are of the same "size", thus both are "mxn" matrices, with the same "m" and "n".
਍匀漀 戀漀琀栀 栀愀瘀攀 琀栀攀 猀愀洀攀 渀甀洀戀攀爀 漀昀 挀漀氀甀洀渀猀 愀渀搀 爀漀眀猀Ⰰ 氀椀欀攀 昀漀爀 攀砀愀洀瀀氀攀 ∀㌀砀㌀∀ 漀爀 ∀㈀砀㐀∀ 漀爀 ∀㈀砀㔀∀ 攀琀挀⸀⸀⸀㰀戀爀㸀ഀഀ
਍匀甀瀀瀀漀猀攀㨀㰀戀爀㸀ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a₁₁ a₁₂ a₁₃ a₁₄ ┐
਍☀⌀㤀㐀㤀㈀㬀 愀㰀猀甀戀㸀㈀㄀㰀⼀猀甀戀㸀 愀㰀猀甀戀㸀㈀㈀㰀⼀猀甀戀㸀 愀㰀猀甀戀㸀㈀㌀㰀⼀猀甀戀㸀 愀㰀猀甀戀㸀㈀㐀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀⼀吀刀㸀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 ☀⌀㤀㔀㔀㬀愀 ☀⌀㤀㔀㔀㬀戀 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ λc λd ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍䄀 㴀ഀഀ

਍ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ B = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ ਍吀栀攀渀㨀㰀戀爀㸀ഀഀ
਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a₁₁+b₁₁ a₁₂+b₁₂ a₁₃+b₁₃ a₁₄+b₁₄ ┐
਍☀⌀㤀㐀㤀㈀㬀 愀㰀猀甀戀㸀㈀㄀㰀⼀猀甀戀㸀⬀戀㰀猀甀戀㸀㈀㄀㰀⼀猀甀戀㸀 愀㰀猀甀戀㸀㈀㈀㰀⼀猀甀戀㸀⬀戀㰀猀甀戀㸀㈀㈀㰀⼀猀甀戀㸀 愀㰀猀甀戀㸀㈀㌀㰀⼀猀甀戀㸀⬀戀㰀猀甀戀㸀㈀㌀㰀⼀猀甀戀㸀 愀㰀猀甀戀㸀㈀㐀㰀⼀猀甀戀㸀⬀戀㰀猀甀戀㸀㈀㐀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀⼀吀刀㸀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 戀㰀猀甀戀㸀㄀㄀㰀⼀猀甀戀㸀 戀㰀猀甀戀㸀㄀㈀㰀⼀猀甀戀㸀 戀㰀猀甀戀㸀㄀㌀㰀⼀猀甀戀㸀 戀㰀猀甀戀㸀㄀㐀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ b21 b22 b23 b24 ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍䄀 ⬀ 䈀 㴀 䈀 ⬀ 䄀㴀ഀഀ

਍ഀഀ ਍㰀栀㌀㸀㌀⸀ 吀栀攀 䤀搀攀渀琀椀琀礀 䴀愀琀爀椀砀㨀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ This special "square" nxn matrix, will have all a_ij=0, except voor those a_ij where i=j. If i=j, then a_ij=1
਍䤀渀 攀昀昀攀挀琀Ⰰ 漀渀氀礀 琀栀攀 搀椀愀最漀渀愀氀 攀氀攀洀攀渀琀猀Ⰰ 昀爀漀洀 琀栀攀 甀瀀瀀攀爀 氀攀昀琀 琀漀 琀栀攀 氀漀眀攀爀 爀椀最栀琀Ⰰ 眀椀氀氀 戀攀 ∀㄀∀⸀㰀戀爀㸀ഀഀ We will see that this matrix, maps a column vector onto itself. In effect, the matrix does "nothing".
਍䠀攀渀挀攀 琀栀攀 渀愀洀攀 ∀䤀搀攀渀琀椀琀礀 洀愀琀爀椀砀∀Ⰰ 漀昀琀攀渀 搀攀渀漀琀攀搀 戀礀 ∀䤀∀⸀㰀戀爀㸀ഀഀ
਍䠀攀爀攀 琀栀攀 ㌀砀㌀ ∀䤀搀攀渀琀椀琀礀 洀愀琀爀椀砀∀㨀㰀戀爀㸀ഀഀ
਍ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ I = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀刀㸀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 ㄀ 　 　 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ 0 1 0 │ ਍☀⌀㤀㐀㤀㈀㬀 　 　 ㄀ ☀⌀㤀㐀㤀㘀㬀ഀഀ

਍㰀戀爀㸀ഀഀ ਍吀栀攀 ㈀砀㈀ ∀䤀搀攀渀琀椀琀礀 洀愀琀爀椀砀∀ 椀猀 琀栀椀猀 漀渀攀㨀㰀戀爀㸀ഀഀ
਍ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ I = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍ഀഀ Let's perform an operation with 3x3 "I", on the column vector "X"
਍㰀戀爀㸀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 ㄀ 　 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 0 1 ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ IX = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ x ┐
਍☀⌀㤀㐀㜀㐀㬀 礀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ z ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 1x+0y+0z ┐
਍☀⌀㤀㐀㜀㐀㬀 　砀⬀㄀礀⬀　稀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ 0x+0y+1z ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ x ┐
਍☀⌀㤀㐀㜀㐀㬀 礀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ z ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍匀漀Ⰰ ∀䤀∀ 洀愀瀀猀 琀栀攀 瘀攀挀琀漀爀 漀渀琀漀 椀琀猀攀氀昀Ⰰ 愀渀搀 琀栀攀 爀攀猀甀氀琀 瘀攀挀琀漀爀 椀猀 琀栀攀 猀愀洀攀 愀猀 琀栀攀 漀爀椀最椀渀愀氀 瘀攀挀琀漀爀⸀㰀戀爀㸀ഀഀ You may wonder why it is neccessary to mention "I" at all. Well, It can help to understand the "inverse" matrix A^-1 of matrix A.
਍㰀戀爀㸀ഀഀ ਍一漀琀攀㨀㰀戀爀㸀ഀഀ In some articles you may also see the "Kronecker delta" notation for the Identity matrix, using the δ_ij symbol.
਍唀猀椀渀最 琀栀椀猀 渀漀琀愀琀椀漀渀Ⰰ 椀琀 椀猀 甀渀搀攀爀猀琀漀漀搀 ⠀漀爀 愀挀琀甀愀氀氀礀 愀最爀攀攀搀 甀瀀漀渀⤀Ⰰ 琀栀愀琀㨀㰀戀爀㸀ഀഀ
਍☀⌀㤀㐀㠀㬀㰀猀甀戀㸀椀樀㰀⼀猀甀戀㸀 㴀 　 昀漀爀 椀 㰀㸀 樀Ⰰ 愀渀搀㰀戀爀㸀ഀഀ δ_ij = 1 for i = j.
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀㐀⸀ 吀栀攀 䤀渀瘀攀爀猀攀 䴀愀琀爀椀砀㨀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ If the matrix "A" can be identified as a "mapping" (or transformation), then generally we may write:
਍ഀഀ

AX = X'

਍ഀഀ In many cases, the mapping A has an "inverse" mapping A^inv, or also notated as A^-1,
਍眀栀椀挀栀 椀猀 琀栀攀 瀀爀攀挀椀猀攀 漀瀀瀀漀猀椀琀攀 洀愀瀀瀀椀渀最 漀昀 䄀⸀㰀戀爀㸀ഀഀ
਍匀甀瀀瀀漀猀攀 琀栀愀琀 䄀 椀猀 愀 挀氀漀挀欀眀椀猀攀 爀漀琀愀琀椀漀渀 漀昀 瘀攀挀琀漀爀猀 椀渀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 漀瘀攀爀 猀漀洀攀 愀渀最氀攀 ☀⌀㤀㠀㄀㬀⸀㰀戀爀㸀ഀഀ Then A^inv is the counter-clockwise rotation over that same angle ϕ.
਍圀栀攀渀 礀漀甀 愀瀀瀀氀礀 䄀Ⰰ 愀渀搀 渀攀砀琀 䄀㰀猀甀瀀㸀椀渀瘀㰀⼀猀甀瀀㸀Ⰰ 愀 瘀攀挀琀漀爀 椀猀 愀最愀椀渀 洀愀瀀瀀攀搀 漀渀琀漀 椀琀猀攀氀昀⸀㰀戀爀㸀ഀഀ
਍䘀漀爀 愀 洀愀瀀瀀椀渀最 䄀 愀渀搀 椀琀✀猀 椀渀瘀攀爀猀攀 洀愀瀀瀀椀渀最 䄀㰀猀甀瀀㸀椀渀瘀㰀⼀猀甀瀀㸀Ⰰ 椀琀 栀漀氀搀猀 琀栀愀琀㨀㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀 䄀䄀㰀猀甀瀀㸀椀渀瘀㰀⼀猀甀瀀㸀堀 㴀 堀㰀⼀栀㌀㸀ഀഀ ਍漀爀㨀㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀 䄀䄀㰀猀甀瀀㸀椀渀瘀㰀⼀猀甀瀀㸀 㴀 䤀㰀⼀栀㌀㸀ഀഀ ਍匀漀 椀昀 礀漀甀 愀瀀瀀氀礀 䄀 ⠀漀渀 猀漀洀攀 猀漀甀爀挀攀 瘀攀挀琀漀爀⤀Ⰰ 愀渀搀 琀栀攀渀 愀瀀瀀氀礀 䄀㰀猀甀瀀㸀椀渀瘀㰀⼀猀甀瀀㸀Ⰰ 琀栀攀渀 琀栀攀 爀攀猀甀氀琀 椀猀 琀栀愀琀 礀漀甀 栀愀瘀攀 琀栀攀 漀爀椀最椀渀愀氀 愀最愀椀渀⸀㰀戀爀㸀ഀഀ
਍ഀഀ ਍㰀栀㌀㸀㔀⸀ 嘀椀攀眀椀渀最 琀栀攀 爀漀眀猀 愀渀搀 挀漀氀甀洀渀猀 漀昀 愀 䴀愀琀爀椀砀Ⰰ 愀猀 瘀攀挀琀漀爀猀㨀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ We have seen quite a few nxm and square nxn matrices by now.
਍㰀戀爀㸀ഀഀ Sometimes, it can be handy to view the "n" columns of the matrix, as "n" columnvectors.
਍䄀渀搀Ⰰ 氀椀欀攀眀椀猀攀Ⰰ 琀漀 瘀椀攀眀 琀栀攀 ∀洀∀ 爀漀眀猀 漀昀 琀栀攀 洀愀琀爀椀砀Ⰰ 愀猀 ∀洀∀ 爀漀眀瘀攀挀琀漀爀猀⸀㰀戀爀㸀ഀഀ
਍吀栀攀爀攀 愀爀攀 猀漀洀攀 最漀漀搀 爀攀愀猀漀渀猀 昀漀爀 瘀椀攀眀椀渀最 愀 洀愀琀爀椀砀 琀栀愀琀 眀愀礀Ⰰ 椀渀 猀漀洀攀 漀挀挀愀猀椀漀渀猀⸀ 䘀漀爀 攀砀愀洀瀀氀攀Ⰰ 猀甀瀀瀀漀猀攀 琀栀愀琀 礀漀甀 猀攀攀㰀戀爀㸀ഀഀ that for a certain matrix, the columnvectors are not independent (like the second column is just equal to a scalar times the first column),
਍琀栀攀渀 琀栀攀 愀爀攀 椀洀瀀氀椀挀愀琀椀漀渀猀 昀漀爀 琀栀攀 ∀洀愀瀀瀀椀渀最∀ 琀栀愀琀 琀栀椀猀 洀愀琀爀椀砀 爀攀瀀爀攀猀攀渀琀猀⸀㰀戀爀㸀ഀഀ
਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a₁₁ a₁₂ a₁₃ ┐
਍☀⌀㤀㐀㜀㐀㬀 愀㰀猀甀戀㸀㈀㄀㰀⼀猀甀戀㸀 愀㰀猀甀戀㸀㈀㈀㰀⼀猀甀戀㸀 愀㰀猀甀戀㸀㈀㌀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ a₃₁ a₃₂ a₃₃ ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍吀栀攀渀 猀漀洀攀琀椀洀攀猀 椀琀✀猀 栀愀渀搀礀 琀漀 琀愀欀攀 愀 挀氀漀猀攀爀 氀漀漀欀 愀琀 琀栀攀 挀漀氀甀洀渀 瘀攀挀琀漀爀猀 漀昀 䄀⸀㰀戀爀㸀ഀഀ
਍䘀漀爀 攀砀愀洀瀀氀攀Ⰰ 愀爀攀 琀栀攀 昀漀氀氀漀眀椀渀最 瘀攀挀琀漀爀猀㨀㰀戀爀㸀ഀഀ
਍ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 ㄀ 　 　 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ 0 1 0 │ ਍☀⌀㤀㐀㤀㈀㬀 　 　 ㄀ ☀⌀㤀㐀㤀㘀㬀ഀഀ	਍㴀ഀഀ	਍㴀ഀഀ
਍䄀 㴀ഀഀ

਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a₁₁ ┐
਍☀⌀㤀㐀㜀㐀㬀 愀㰀猀甀戀㸀㈀㄀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ a₃₁ ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍ഀഀ ਍㰀吀䐀㸀ഀഀ ,   ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a₁₃ ┐
਍☀⌀㤀㐀㜀㐀㬀 愀㰀猀甀戀㸀㈀㌀㰀⼀猀甀戀㸀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ a₃₃ ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀刀㸀ഀഀ

਍Ⰰ ☀渀戀猀瀀㬀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 愀㰀猀甀戀㸀㄀㈀㰀⼀猀甀戀㸀  ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ a22 │ ਍☀⌀㤀㐀㤀㈀㬀 愀㰀猀甀戀㸀㌀㈀㰀⼀猀甀戀㸀  ☀⌀㤀㐀㤀㘀㬀ഀഀ

਍㰀戀爀㸀ഀഀ independent? That is, is one vector not just a "multiplication" of a number with another vector?
਍㰀戀爀㸀ഀഀ In chapter 3, we will appreciate this better.
਍䘀漀爀 渀漀眀Ⰰ 琀栀攀 漀渀氀礀 琀栀椀渀最 琀栀愀琀 䤀 氀椀欀攀 琀漀 猀愀礀 栀攀爀攀Ⰰ 椀猀 琀栀愀琀 眀攀 挀愀渀 愀氀猀漀 瘀椀攀眀 愀 洀愀琀爀椀砀 愀猀 愀 猀攀琀㰀戀爀㸀ഀഀ columnvectors, or rowvectors, which may give us some extra insights on several occasions.
਍ഀഀ ਍ഀഀ ਍㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀㘀⸀ 吀栀攀 吀爀愀渀猀瀀漀猀攀 漀昀 愀 䴀愀琀爀椀砀㨀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ If you have any mxn matrix A, then if you simply interchange all the rows and columns,
਍礀漀甀 最攀琀 琀栀攀 琀爀愀渀猀瀀漀猀攀 洀愀琀爀椀砀 㰀猀甀瀀㸀琀㰀⼀猀甀瀀㸀䄀⸀㰀戀爀㸀ഀഀ
਍䔀砀愀洀瀀氀攀㨀㰀戀爀㸀ഀഀ
਍ഀഀ Suppose that matrix A is:
਍㰀戀爀㸀ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍䄀 㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 2 1 0 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀ ㌀ 㔀 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀⼀吀刀㸀ഀഀ ਍ഀഀ Then the transpose Matrix "^tA" is:
਍ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍㰀猀甀瀀㸀琀㰀⼀猀甀瀀㸀䄀 㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 2 1 ┐
਍☀⌀㤀㐀㜀㐀㬀 ㄀ ㌀ ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ 0 5 ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍吀栀攀爀攀 愀爀攀 猀攀瘀攀爀愀氀 爀攀愀猀漀渀猀 眀栀礀 琀栀攀 琀爀愀渀猀瀀漀猀攀 洀愀琀爀椀砀 椀猀 搀攀昀椀渀攀搀⸀ 伀渀攀 漀昀 眀栀椀挀栀 眀攀 眀椀氀氀 猀攀攀 氀愀琀攀爀 漀渀⸀㰀戀爀㸀ഀഀ
਍伀渀攀 爀攀愀猀漀渀 眀攀 眀椀氀氀 猀攀攀 栀攀爀攀 渀漀眀㨀 椀昀 愀 渀砀渀 洀愀琀爀椀砀 椀猀 ∀猀礀洀洀攀琀爀椀挀∀ 眀椀琀栀 爀攀猀瀀攀挀琀 琀漀 椀琀✀猀 搀椀愀最漀渀愀氀Ⰰ 琀栀攀渀 䄀 㴀 㰀猀甀瀀㸀琀㰀⼀猀甀瀀㸀䄀Ⰰ㰀戀爀㸀ഀഀ thus in that case, the matrix A is fully equal to it's transpose matrix.
਍㰀戀爀㸀ഀഀ Example of a symmetric matrix (neccessarily nxn ofcourse):
਍㰀戀爀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ A = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀刀㸀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 ㈀ ㄀ 㔀☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ 1 3 0│ ਍☀⌀㤀㐀㤀㈀㬀 㔀 　 ㈀☀⌀㤀㐀㤀㘀㬀ഀഀ

਍㰀戀爀㸀ഀഀ For that matrix, it is true that A = ^tA. You can easily check it by interchanging the rows and columns.
਍㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀㜀⸀ 吀栀攀 䐀攀琀攀爀洀椀渀愀渀琀 漀昀 愀 䴀愀琀爀椀砀㨀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ For a square nxn matrix, you can calculate it's determinant. It's just a scalar, specific for that matrix,
਍甀猀椀渀最 愀 挀攀爀琀愀椀渀 挀愀氀挀甀氀愀琀椀漀渀 漀昀 琀栀攀 攀氀攀洀攀渀琀猀 愀㰀猀甀戀㸀椀樀㰀⼀猀甀戀㸀 漀昀 琀栀愀琀 洀愀琀爀椀砀⸀㰀戀爀㸀ഀഀ
਍吀栀椀猀 渀甀洀戀攀爀Ⰰ 戀攀氀漀渀最椀渀最 琀漀 琀栀攀 洀愀琀爀椀砀 㰀䈀㸀䄀㰀⼀䈀㸀Ⰰ 椀猀 搀攀渀漀琀攀搀 戀礀 攀椀琀栀攀爀 㰀䈀㸀搀攀琀⠀䄀⤀㰀⼀䈀㸀Ⰰ 愀渀搀 漀昀琀攀渀 愀氀猀漀 戀礀 㰀䈀㸀簀䄀簀㰀⼀䈀㸀⸀㰀戀爀㸀ഀഀ
਍䘀漀爀 愀 ㈀砀㈀ 洀愀琀爀椀砀Ⰰ 琀栀攀 搀攀琀攀爀洀椀渀愀渀琀 椀猀 挀愀氀挀甀氀愀琀攀搀 愀猀 椀渀 琀栀攀 攀砀愀洀瀀氀攀 戀攀氀漀眀㨀㰀戀爀㸀ഀഀ
਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍䄀 㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a b ┐
਍☀⌀㤀㐀㤀㈀㬀 挀 搀 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀⼀吀刀㸀ഀഀ ਍㰀戀爀㸀ഀഀ Then:
਍ഀഀ

det(A) = ad - bc

਍ഀഀ What use can we possibly have from calculating the determinant?
਍㰀戀爀㸀ഀഀ There are several properties of the matrix you can immediately deduce, once you have the determinant.
਍伀渀攀 椀洀瀀漀爀琀愀渀琀 漀渀攀 椀猀 琀栀椀猀㨀 挀愀渀 眀攀 瘀椀攀眀 琀栀攀 挀漀氀甀洀渀瘀攀挀琀漀爀猀 漀昀 琀栀攀 洀愀琀爀椀砀 愀猀 戀攀椀渀最 㰀䈀㸀椀渀搀攀瀀攀渀搀攀渀琀㰀⼀䈀㸀㼀㰀戀爀㸀ഀഀ
਍䌀漀渀猀椀搀攀爀 琀栀椀猀 洀愀琀爀椀砀㨀㰀戀爀㸀ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍䄀 㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 3 0 ┐
਍☀⌀㤀㐀㤀㈀㬀 　 ㌀ ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀⼀吀刀㸀ഀഀ ਍㰀戀爀㸀ഀഀ Then |A| = 3*3 - 0*0 = 9.
਍㰀戀爀㸀ഀഀ Now, consider this matrix:
਍ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ B = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍吀栀攀渀 簀䈀簀 㴀 ㄀⨀㐀 ⴀ ㈀⨀㈀ 㴀 　⸀㰀戀爀㸀ഀഀ
਍䤀渀 琀栀攀 氀愀琀琀攀爀 挀愀猀攀Ⰰ 簀䈀簀㴀　⸀ 䤀昀 礀漀甀 氀漀漀欀 洀漀爀攀 挀氀漀猀攀氀礀 琀漀 琀栀攀 挀漀氀甀洀渀瘀攀挀琀漀爀猀 漀昀 洀愀琀爀椀砀 䈀Ⰰ㰀戀爀㸀ഀഀ then those two vectors are not independent.
਍䠀攀爀攀 眀攀 挀愀渀 猀攀攀 琀栀愀琀㨀㰀戀爀㸀ഀഀ
਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍㰀吀䐀㸀ഀഀ 2* ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍匀漀Ⰰ 琀栀漀猀攀 琀眀漀 挀漀氀甀洀渀瘀攀挀琀漀爀猀 愀爀攀 渀漀琀 椀渀搀攀瀀攀渀搀攀渀琀⸀㰀戀爀㸀ഀഀ If the det(A)=0 for some matrix A, then we know that the columnvectors are not all independent,
਍眀栀椀挀栀 挀愀渀 戀攀 椀洀瀀漀爀琀愀渀琀 椀渀 挀攀爀琀愀椀渀 漀挀挀愀猀椀漀渀猀 ⠀猀攀攀 挀栀愀瀀琀攀爀 ㌀⤀⸀㰀戀爀㸀ഀഀ ਍㰀戀爀㸀ഀഀ
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀甀攀∀㸀ഀഀ

Chapter 3. More on Linear mappings or Linear transformations:

਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍䤀琀✀猀 琀爀甀攀 琀栀愀琀 洀愀渀礀 猀甀戀樀攀挀琀猀 椀渀 瀀栀礀猀椀挀猀Ⰰ 挀愀渀 戀攀 搀攀猀挀爀椀戀攀搀 戀礀 眀栀愀琀 椀猀 挀愀氀氀攀搀 ∀愀 氀椀渀攀愀爀 洀愀瀀瀀椀渀最∀⸀㰀戀爀㸀ഀഀ
਍圀攀 愀氀爀攀愀搀礀 栀愀瘀攀 猀攀攀渀 猀漀洀攀 攀砀愀洀瀀氀攀猀Ⰰ 氀椀欀攀 琀栀攀 爀漀琀愀琀椀漀渀 漀昀 瘀攀挀琀漀爀猀 椀渀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 漀瘀攀爀 愀渀 愀渀最氀攀Ⰰ 眀栀椀挀栀 挀愀渀㰀戀爀㸀ഀഀ can be described by a matrix.
਍㰀戀爀㸀ഀഀ However, there also exist several classes of "non-linear mappings", which often are a subject in
਍洀漀爀攀 愀搀瘀愀渀挀攀搀 渀漀琀攀猀⸀㰀戀爀㸀ഀഀ
਍吀栀攀 搀攀昀椀渀椀琀椀漀渀 漀昀 愀 䰀椀渀攀愀爀 洀愀瀀瀀椀渀最 漀爀 䰀椀渀攀愀爀 琀爀愀渀猀昀漀爀洀愀琀椀漀渀 椀猀 挀攀爀琀愀椀渀氀礀 渀漀琀 搀椀昀昀椀挀甀氀琀⸀㰀戀爀㸀ഀഀ We will see it in a moment.
਍㰀戀爀㸀ഀഀ In a somewhat more formal description in linear algebra, folks speak of vectorspaces "V" and "W",
਍眀栀攀爀攀 戀漀琀栀 挀愀渀 栀愀瘀攀 㰀䤀㸀愀渀礀 渀甀洀戀攀爀 漀昀 搀椀洀攀渀猀椀漀渀猀⸀㰀⼀䤀㸀㰀戀爀㸀ഀഀ
਍圀攀 漀昀琀攀渀Ⰰ ∀猀椀氀攀渀琀氀礀∀Ⰰ 愀猀猀甀洀攀搀 琀栀愀琀 眀攀 眀攀爀攀 搀攀愀氀椀渀最 眀椀琀栀 愀 洀愀瀀瀀椀渀最 昀爀漀洀 攀⸀最⸀ 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 ⴀ㸀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀Ⰰ 漀爀 刀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀 ⴀ㸀 刀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀 攀琀挀⸀⸀㰀戀爀㸀ഀഀ However, exploring mappings from e.g. R⁴ -> R², are very common subjects in linear algebra.
਍㰀戀爀㸀ഀഀ So, when a formal description talks of vectorspaces "V" and "W", you ofcourse may think of examples like R³ -> R³,
਍漀爀 猀椀洀瀀氀礀 最漀 眀椀琀栀 琀栀攀 最攀渀攀爀愀氀 昀漀爀洀甀氀愀琀椀漀渀 甀猀椀渀最 琀栀攀 嘀 愀渀搀 圀 瘀攀挀琀漀爀猀瀀愀挀攀猀⸀㰀戀爀㸀ഀഀ
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀甀攀∀㸀ഀഀ

3.1 What is a Linear Transformation?

਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍䰀攀琀 嘀 愀渀搀 圀 戀攀 瘀攀挀琀漀爀猀瀀愀挀攀猀 漀瘀攀爀 琀栀攀 昀椀攀氀搀 䬀⸀㰀戀爀㸀ഀഀ Let the vectors u and v be members of V (that is: u and v ∈ V).
਍䰀攀琀 ☀⌀㤀㔀㔀㬀 戀攀 愀渀 攀氀攀洀攀渀琀 漀昀 䬀⸀㰀戀爀㸀ഀഀ
਍䄀 㰀䈀㸀氀椀渀攀愀爀 洀愀瀀瀀椀渀最㰀⼀䈀㸀 䘀㨀 嘀 ⴀ㸀 圀 椀猀 愀 洀愀瀀瀀椀渀最 眀椀琀栀 琀栀攀 瀀爀漀瀀攀爀琀椀攀猀㨀㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀 䘀⠀甀 ⬀ 瘀⤀ 㴀 䘀⠀甀⤀ ⬀ 䘀⠀瘀⤀㰀⼀栀㌀㸀ഀഀ

F(λv) = λ F(v)

਍ഀഀ Note that V -> W might thus also be V -> V (like R³ -> R³).
਍㰀戀爀㸀ഀഀ ਍吀栀攀猀攀 愀爀攀 瀀爀攀琀琀礀 猀椀洀瀀氀攀 爀甀氀攀猀⸀ 䤀渀 愀 椀渀琀攀爀瀀爀攀琀愀琀椀漀渀 昀爀漀洀 瀀栀礀猀椀挀猀Ⰰ 漀渀攀 洀椀最栀琀 猀愀礀 琀栀愀琀 琀栀攀 洀愀瀀瀀椀渀最 栀愀猀 渀漀 戀椀愀猀㰀戀爀㸀ഀഀ for a certain direction in the vectorspaces.
਍夀漀甀 洀椀最栀琀 愀氀猀漀 猀愀礀 琀栀愀琀 礀漀甀 猀攀攀 猀漀洀攀 ∀愀猀猀漀挀椀愀琀椀瘀攀∀ 愀渀搀 ∀搀椀猀琀爀椀戀甀琀椀瘀攀∀ 瀀爀漀瀀攀爀琀椀攀猀 栀攀爀攀⸀㰀戀爀㸀ഀഀ
਍䄀猀 愀渀 攀砀愀洀瀀氀攀 漀昀 瀀爀漀瀀攀爀琀礀 ㄀㨀㰀戀爀㸀ഀഀ
਍匀甀瀀瀀漀猀攀 眀攀 栀愀瘀攀 愀 爀漀琀愀琀椀漀渀 漀瀀攀爀愀琀漀爀 䄀⸀㰀戀爀㸀ഀഀ
਍㰀䈀㸀䴀攀琀栀漀搀 ㄀㨀㰀⼀䈀㸀㰀戀爀㸀ഀഀ Let's first add two vectors u and v in the usual way. This will yield result vector w = u + v.
਍一攀砀琀Ⰰ 眀攀 氀攀琀 琀栀攀 爀漀琀愀琀椀漀渀 漀瀀攀爀愀琀漀爀 眀漀爀欀 漀渀 㰀䈀㸀眀㰀⼀䈀㸀⸀ 吀栀椀猀 眀椀氀氀 礀椀攀氀搀 㰀䈀㸀眀✀㰀⼀䈀㸀⸀㰀戀爀㸀ഀഀ
਍圀椀氀氀 琀栀椀猀 戀攀 琀栀攀 猀愀洀攀 愀猀 琀栀攀 昀漀氀氀漀眀椀渀最 猀攀焀甀攀渀挀攀 漀昀 攀瘀攀渀琀猀㼀㰀戀爀㸀ഀഀ
਍㰀䈀㸀䴀攀琀栀漀搀 ㈀㨀㰀⼀䈀㸀㰀戀爀㸀ഀഀ Let the rotation operator first work on u and v. This will yield the result vectors u' and v'.
਍吀栀攀渀 眀攀 眀椀氀氀 愀搀搀 㰀䈀㸀甀✀㰀⼀䈀㸀 愀渀搀 㰀䈀㸀瘀✀㰀⼀䈀㸀Ⰰ 爀攀猀甀氀琀椀渀最 椀渀 㰀䈀㸀眀✀㰀⼀䈀㸀 㴀 㰀䈀㸀甀✀㰀⼀䈀㸀 ⬀ 㰀䈀㸀瘀✀㰀⼀䈀㸀⸀㰀戀爀㸀ഀഀ
਍䐀漀攀猀 戀漀琀栀 洀攀琀栀漀搀猀 爀攀猀甀氀琀 椀渀 琀栀攀 猀愀洀攀 㰀䈀㸀眀✀㰀⼀䈀㸀 㼀 䤀昀 猀漀Ⰰ 琀栀攀渀 䄀 椀猀 愀 氀椀渀攀愀爀 洀愀瀀瀀椀渀最⸀㰀戀爀㸀ഀഀ
਍㰀栀㌀㸀䰀攀琀✀猀 琀爀礀 椀琀Ⰰ 甀猀椀渀最 愀 猀椀洀瀀氀攀 攀砀愀洀瀀氀攀⸀㰀⼀栀㌀㸀ഀഀ ਍䰀攀琀 琀栀攀 洀愀瀀瀀椀渀最 䄀 栀愀瘀攀 琀栀攀 愀猀猀漀挀椀愀琀攀 洀愀琀爀椀砀㨀㰀戀爀㸀ഀഀ
਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 0 -1 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀⸀ 　 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀⼀吀刀㸀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 ㄀ ㈀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 2 4 ┘ ਍㰀⼀吀䐀㸀ഀഀ
਍☀⌀㤀㐀㠀㐀㬀 ㈀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 4 ┘ ਍㰀⼀吀䐀㸀ഀഀ	਍㴀ഀഀ	਍☀⌀㤀㐀㠀㐀㬀 ㄀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 2 ┘ ਍㰀⼀吀䐀㸀ഀഀ
਍䄀 㴀ഀഀ

਍ഀഀ This is a counter clockwise rotation over 90 degrees. Yes, I indeed choose a simple one, but that does not matter at all.
਍㰀戀爀㸀ഀഀ Let the vectors u and v be: ਍ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ u = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 0 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀ ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀⼀吀刀㸀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 ㄀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 0 ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍瘀 㴀ഀഀ

਍ഀഀ Yes, these are the standard basis vectors of R², but that's OK too.
਍㰀戀爀㸀ഀഀ => Let's do method 1 first.
਍㰀戀爀㸀ഀഀ Adding u and v results in the vector w = (1,1).
਍一攀砀琀Ⰰ 氀攀琀 䄀 漀瀀攀爀愀琀攀 漀渀 㰀䈀㸀眀㰀⼀䈀㸀㨀㰀戀爀㸀ഀഀ
਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍䄀眀 㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 0 -1 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀⸀ 　 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 1 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀ ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀ഀഀ = ਍㰀⼀吀䐀㸀ഀഀ ਍☀⌀㤀㐀㠀㐀㬀 ⴀ㄀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 1. ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀刀㸀ഀഀ ਍㰀戀爀㸀ഀഀ ਍㰀䈀㸀㴀㸀 一攀砀琀 眀攀 搀漀 洀攀琀栀漀搀 ㈀㨀㰀⼀䈀㸀㰀戀爀㸀ഀഀ
਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍䄀甀 㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 0 -1 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀⸀ 　 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 1 ┐
਍☀⌀㤀㐀㤀㈀㬀 　 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀ഀഀ = ਍㰀⼀吀䐀㸀ഀഀ ਍☀⌀㤀㐀㠀㐀㬀 　 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 1 ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀刀㸀ഀഀ ਍ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ Av = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ -1 ┐
਍☀⌀㤀㐀㤀㈀㬀 　⸀ ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀 ഀഀ ਍㰀戀爀㸀ഀഀ So:
਍ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 　 ⴀ㄀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 1. 0 ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 　 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 1 ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍㴀ഀഀ

਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ Au + Av = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 0 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀ ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀ഀഀ + ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ -1 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀⸀ ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀 ഀഀ
਍夀攀猀Ⰰ 戀漀琀栀 洀攀琀栀漀搀猀 眀漀爀欀 漀甀琀 琀栀攀 猀愀洀攀 眀愀礀⸀ 匀漀Ⰰ 椀渀 琀栀椀猀 瀀愀爀琀椀挀甀氀愀爀 攀砀愀洀瀀氀攀 眀攀 栀愀瘀攀 㰀䈀㸀䄀⠀甀⬀瘀⤀ 㴀 䄀⠀甀⤀ ⬀ 䄀⠀瘀⤀㰀⼀䈀㸀㰀戀爀㸀ഀഀ
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀甀攀∀㸀ഀഀ

3.2 A Linear Transformation and it's Matrix, and the mapping of the basis vectors.

਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍㰀栀㌀㸀䔀砀愀洀瀀氀攀 漀昀 搀椀爀攀挀琀氀礀 昀椀渀搀椀渀最 琀栀攀 洀愀琀爀椀砀 漀昀 猀漀洀攀 䰀椀渀攀愀爀 吀爀愀渀猀昀漀爀洀愀琀椀漀渀㨀㰀⼀栀㌀㸀ഀഀ ਍匀甀瀀瀀漀猀攀 眀攀 栀愀瘀攀 愀 氀椀渀攀愀爀 洀愀瀀瀀椀渀最 㰀䈀㸀䘀㨀㰀⼀䈀㸀 刀㰀猀甀瀀㸀㐀㰀⼀猀甀瀀㸀 ⴀ㸀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀⸀㰀戀爀㸀ഀഀ
਍夀攀猀Ⰰ 琀栀椀猀 椀猀 昀爀漀洀 愀 㐀 搀椀洀攀渀猀椀漀渀愀氀 瘀攀挀琀漀爀猀瀀愀挀攀 ⠀嘀⤀ 琀漀 漀甀爀 昀愀洀椀氀椀愀爀 ㈀ 搀椀洀攀渀猀椀漀渀愀氀 瘀攀挀琀漀爀猀瀀愀挀攀 ⠀圀⤀⸀㰀戀爀㸀ഀഀ
਍匀甀瀀瀀漀猀攀 昀甀爀琀栀攀爀 琀栀愀琀 琀栀攀 猀攀琀 漀昀 瘀攀挀琀漀爀猀 笀䔀㰀猀甀瀀㸀㄀㰀⼀猀甀瀀㸀Ⰰ 䔀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀Ⰰ 䔀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀Ⰰ 䔀㰀猀甀瀀㸀㐀㰀⼀猀甀瀀㸀紀 昀漀爀洀 愀 猀攀琀 漀昀 戀愀猀椀猀瘀攀挀琀漀爀猀 椀渀 嘀⸀㰀戀爀㸀ഀഀ
਍一攀砀琀Ⰰ 猀甀瀀瀀漀猀攀 琀栀愀琀 眀攀 欀渀漀眀 漀昀 琀栀攀 昀漀氀氀漀眀椀渀最 洀愀瀀瀀椀渀最猀 漀昀 㰀䈀㸀䘀㰀⼀䈀㸀㨀㰀戀爀㸀ഀഀ
਍ഀഀ

਍䄀眀 㴀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 ⴀ㄀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ └ 0. ┘ ਍㰀⼀吀䐀㸀ഀഀ

਍㴀ഀഀ

਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍䘀⠀䔀㰀猀甀瀀㸀㄀㰀⼀猀甀瀀㸀⤀㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 2 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀ ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀ഀഀ ,   ਍㰀⼀吀䐀㸀ഀഀ ਍䘀⠀䔀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀⤀㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 3. ┐
਍☀⌀㤀㐀㤀㈀㬀 ⴀ㄀ ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀ഀഀ ,   ਍㰀⼀吀䐀㸀ഀഀ ਍䘀⠀䔀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀⤀㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ -5 ┐
਍☀⌀㤀㐀㤀㈀㬀 㐀⸀ ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀吀䐀㸀ഀഀ ,   ਍㰀⼀吀䐀㸀ഀഀ ਍䘀⠀䔀㰀猀甀瀀㸀㐀㰀⼀猀甀瀀㸀⤀㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 1 ┐
਍☀⌀㤀㐀㤀㈀㬀 㜀 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀⼀吀刀㸀ഀഀ ਍㰀戀爀㸀ഀഀ Then I immediately know the matrix that we can associate with F, namely
਍㰀戀爀㸀ഀഀ ਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍䘀 㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 2 3. -5 1 ┐
਍☀⌀㤀㐀㤀㈀㬀 ㄀ ⴀ㄀  㐀⸀ 㜀 ☀⌀㤀㐀㤀㘀㬀ഀഀ ਍㰀⼀吀刀㸀ഀഀ ਍㰀戀爀㸀ഀഀ I will make it plausible to you, how I came to this knowledge.
਍㰀戀爀㸀ഀഀ I will show you, that if you have a linear mapping "F", that then the mappings of the basisvectors, immediately
਍眀椀氀氀 猀栀漀眀 礀漀甀 琀栀攀 挀漀氀甀洀渀 瘀攀挀琀漀爀猀 漀昀 琀栀攀 洀愀琀爀椀砀 漀昀 ∀䘀∀⸀㰀⼀䤀㸀㰀⼀䈀㸀㰀戀爀㸀ഀഀ
਍䤀渀 昀愀挀琀Ⰰ 琀栀椀猀 椀猀 攀砀愀挀琀氀礀 眀栀愀琀 栀愀瀀瀀攀渀攀搀 椀渀 琀栀攀 攀砀愀洀瀀氀攀 愀戀漀瘀攀⸀㰀戀爀㸀ഀഀ Let's try this in R³. Watch the following reasoning.
਍㰀戀爀㸀ഀഀ Suppose F is a "linear mapping" from R³ to R³.
਍匀漀Ⰰ 眀攀 猀甀瀀瀀漀猀攀 琀栀愀琀 琀栀攀 洀愀琀爀椀砀 漀昀 䘀 椀猀㨀㰀戀爀㸀ഀഀ
਍㰀吀䄀䈀䰀䔀 戀漀爀搀攀爀㴀　㸀 ഀഀ ਍䘀 㴀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a b c ┐
਍☀⌀㤀㐀㜀㐀㬀 搀 攀 昀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ g h i ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍圀攀 昀甀爀琀栀攀爀 搀漀 渀漀琀 欀渀漀眀 愀渀礀琀栀椀渀最 漀昀 䘀Ⰰ 攀砀挀攀瀀琀 琀栀愀琀 椀琀 椀猀 愀 氀椀渀攀愀爀 洀愀瀀瀀椀渀最⸀ 䠀攀渀挀攀 愀氀猀漀 琀栀攀 甀渀欀渀漀眀 攀氀攀洀攀渀琀猀 椀渀 琀栀攀 洀愀琀爀椀砀 愀戀漀瘀攀⸀㰀戀爀㸀ഀഀ
਍䤀渀 刀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀Ⰰ 眀攀 栀愀瘀攀 琀栀攀 昀漀氀氀漀眀椀渀最 猀攀琀 漀昀 漀爀琀栀漀渀漀爀洀愀氀 戀愀猀椀猀瘀攀挀琀漀爀猀㨀㰀戀爀㸀ഀഀ
਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ 1 ┐
਍☀⌀㤀㐀㜀㐀㬀 　 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ 0 ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍ഀഀ ਍ഀഀ ਍ഀഀ ਍ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀 ഀഀ
਍一攀砀琀Ⰰ 眀攀 氀攀琀 琀栀攀 洀愀琀爀椀砀 䘀 漀瀀攀爀愀琀攀 漀渀 漀甀爀 戀愀猀椀猀瘀攀挀琀漀爀猀⸀ 䤀 眀椀氀氀 搀漀 琀栀椀猀 漀渀氀礀 昀漀爀 琀栀攀 ⠀㄀Ⰰ　Ⰰ　⤀ 戀愀猀椀猀瘀攀挀琀漀爀⸀㰀戀爀㸀ഀഀ For the other two, the same principle applies. So, this will yield:
਍㰀戀爀㸀ഀഀ

਍Ⰰ ☀渀戀猀瀀㬀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 　 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ 1 │ ਍☀⌀㤀㐀㤀㈀㬀 　 ☀⌀㤀㐀㤀㘀㬀ഀഀ

਍Ⰰ ☀渀戀猀瀀㬀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 　 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ 0 │ ਍☀⌀㤀㐀㤀㈀㬀 ㄀ ☀⌀㤀㐀㤀㘀㬀ഀഀ

਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ a b c ┐
਍☀⌀㤀㐀㜀㐀㬀 搀 攀 昀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ g h i ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀ഀഀ = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀ഀഀ = ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀刀㸀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 ㄀ ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ 0 │ ਍☀⌀㤀㐀㤀㈀㬀 　 ☀⌀㤀㐀㤀㘀㬀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 愀⨀㄀⬀戀⨀　⬀挀⨀　 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ d*1+e*0+f*0 │ ਍☀⌀㤀㐀㤀㈀㬀 最⨀㄀⬀栀⨀　⬀椀⨀　 ☀⌀㤀㐀㤀㘀㬀ഀഀ

਍☀⌀㤀㐀㠀㐀㬀 愀 ☀⌀㤀㐀㠀㠀㬀㰀戀爀㸀ഀഀ │ d │ ਍☀⌀㤀㐀㤀㈀㬀 最 ☀⌀㤀㐀㤀㘀㬀ഀഀ

਍㰀戀爀㸀ഀഀ Well, this is indeed the first column vector of the matrix F.
਍㰀戀爀㸀ഀഀ So, one fact to remember is: the mappings of the basis vectors correspond to the column vectors of the matrix.
਍㰀戀爀㸀ഀഀ
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀甀攀∀㸀ഀഀ

Chapter 4. Distances in Rⁿ (flat Eucledian space):

਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍㰀䤀㸀䤀渀 愀渀漀琀栀攀爀 渀漀琀攀Ⰰ 䤀 眀椀氀氀 猀栀漀眀 栀漀攀 琀漀 搀攀愀氀 眀椀琀栀 ∀搀椀猀琀愀渀挀攀猀∀ 椀渀 渀漀渀ⴀ䔀甀挀氀攀搀椀愀渀 挀甀爀瘀攀搀 猀瀀愀挀攀猀Ⰰ 漀爀 甀猀椀渀最 愀 搀椀昀昀攀爀攀渀琀 渀漀渀ⴀ䌀愀爀琀攀猀椀愀渀㰀戀爀㸀ഀഀ coordinate system. Then also "contravariant" and "covariant" vectors and indices will be discussed.
਍䤀 琀栀椀渀欀 琀栀愀琀 琀栀椀猀 猀漀爀琀 漀昀 猀琀甀昀昀 眀椀氀氀 最漀 椀渀琀漀 渀漀琀攀 ㄀㠀⸀㰀⼀䤀㸀㰀戀爀㸀ഀഀ
਍䤀渀 琀栀椀猀 猀攀挀琀椀漀渀Ⰰ 眀攀 最漀椀渀最 琀漀 搀椀猀挀甀猀猀 琀栀攀 ∀搀椀猀琀愀渀挀攀∀ 戀攀琀眀攀攀渀 瀀漀椀渀琀猀 椀渀 刀㰀猀甀瀀㸀渀㰀⼀猀甀瀀㸀 ⠀氀椀欀攀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 漀爀 刀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀⤀⸀㰀戀爀㸀ഀഀ
਍夀漀甀 挀愀渀 瘀椀攀眀 琀栀椀猀 猀琀甀昀昀 椀渀 洀甀氀琀椀瀀氀攀 眀愀礀猀⸀ 夀漀甀 挀愀渀 猀愀礀 琀栀愀琀 眀攀 ∀樀甀猀琀∀ 栀愀瘀攀 琀眀漀 瀀漀椀渀琀猀 倀 愀渀搀 倀✀Ⰰ 愀渀搀 琀栀攀 氀椀渀攀 猀攀最洀攀渀琀㰀戀爀㸀ഀഀ which connects those two points, clearly determines the distance between them.
਍夀漀甀 挀愀渀 愀氀猀漀 猀愀礀 琀栀愀琀 眀攀 栀愀瘀攀 愀 瘀攀挀琀漀爀 倀 愀渀搀 瘀攀挀琀漀爀 倀✀Ⰰ 愀渀搀 琀栀攀 氀攀渀最琀栀 漀昀 琀栀攀 瘀攀挀琀漀爀 倀ⴀ倀✀Ⰰ㰀戀爀㸀ഀഀ simply is the distance.
਍䘀漀爀 愀戀漀甀琀 琀栀攀 氀愀猀琀 猀琀愀琀攀洀攀渀琀㨀 琀愀欀攀 愀 氀漀漀欀 愀琀 昀椀最甀爀攀 㐀Ⰰ 琀栀攀 琀栀椀爀搀 瀀椀挀琀甀爀攀⸀ 䠀攀爀攀 眀攀 栀愀瘀攀 瘀攀挀琀漀爀猀 䄀 愀渀搀 䈀Ⰰ 愀渀搀 㰀䈀㸀琀栀攀 氀攀渀最栀琀㰀⼀䈀㸀㰀戀爀㸀ഀഀ of the vector "A-B", defines the distance between the endpoints of A and B.
਍ഀഀ
਍㰀䈀㸀䘀椀最甀爀攀 㐀⸀ 匀漀洀攀 攀砀愀洀瀀氀攀 搀椀猀琀愀渀挀攀猀⸀㰀⼀䈀㸀㰀戀爀㸀ഀഀ
਍㰀椀洀最 猀爀挀㴀∀瘀瀀愀爀琀㈀开㘀⸀樀瀀最∀ 愀氀椀最渀㴀∀挀攀渀琀爀攀∀⼀㸀ഀഀ
਍㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀䤀渀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀㨀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ In figure 4, in the first picture, we have R² space. You can see that we have two points:
਍㰀戀爀㸀ഀഀ P : (x₁, y₁)
਍倀✀㨀 ⠀砀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀Ⰰ 礀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀⤀ ☀渀戀猀瀀㬀 ⠀渀漀琀攀 琀栀攀 愀挀挀攀渀琀 漀渀 琀栀攀 ∀倀∀⤀㰀戀爀㸀ഀഀ
਍圀攀 眀愀渀琀 琀漀 昀椀渀搀 琀栀攀 氀攀渀最琀栀 漀昀 琀栀攀 猀攀最洀攀渀琀 倀倀✀⸀ 唀猀甀愀氀氀礀Ⰰ 琀栀攀 氀攀渀最栀琀 漀昀 猀甀挀栀 愀 猀攀最洀攀渀琀 椀猀 搀攀渀漀琀攀搀 戀礀 簀倀倀✀簀⸀㰀戀爀㸀ഀഀ
਍夀漀甀 挀愀渀 愀氀眀愀礀猀 ∀昀椀渀搀∀ 漀爀 挀漀渀猀琀爀甀挀琀 愀 ∀爀椀最栀琀ⴀ愀渀最氀攀搀 琀爀椀愀渀最氀攀∀Ⰰ 眀栀攀渀 礀漀甀 栀愀瘀攀 琀眀漀 瀀漀椀渀琀猀 椀渀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀⸀㰀戀爀㸀ഀഀ
਍䠀攀爀攀 眀攀 挀愀渀 猀攀攀 琀栀愀琀 琀栀攀 氀攀渀最栀琀 瀀愀爀愀氀氀攀氀 琀栀攀 砀ⴀ愀砀椀猀Ⰰ 椀猀 ∀砀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀ⴀ砀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀∀⸀㰀戀爀㸀ഀഀ And that the lenght parallel the y-axis, is "y₂-y₁".
਍㰀戀爀㸀ഀഀ In our specific example in figure 4, we have x₂-x₁ = 5 - 2 = 3.
਍䄀渀搀 昀漀爀 琀栀攀 礀 瀀愀爀琀Ⰰ 眀攀 栀愀瘀攀 礀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀ⴀ礀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀 㴀 㜀 ⴀ ㌀ 㴀 㐀⸀㰀戀爀㸀ഀഀ
਍䄀瀀瀀氀礀椀渀最 琀栀攀 ∀倀礀琀栀愀最漀爀攀愀渀 琀栀攀漀爀攀洀∀Ⰰ 眀攀 昀椀渀搀㨀㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀簀倀倀✀簀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 㴀 ⠀砀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 砀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 ⬀ ⠀礀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 礀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀㰀⼀栀㌀㸀ഀഀ ਍吀栀甀猀㨀㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀簀倀倀✀簀 㴀 ☀⌀㠀㜀㌀　㬀 ⠀ ⠀砀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 砀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 ⬀ ⠀礀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 礀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀⤀ ⤀㰀⼀栀㌀㸀ഀഀ ਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀㸀ഀഀ

In R³:

਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍䘀漀爀 刀㰀猀甀瀀㸀㌀㰀⼀猀甀瀀㸀Ⰰ 眀攀 猀椀洀瀀氀礀 栀愀瘀攀 漀渀攀 愀搀搀椀琀椀漀渀愀氀 挀漀漀爀搀椀渀愀琀攀⸀ 匀漀Ⰰ 眀攀 眀漀甀氀搀 栀愀瘀攀㨀㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀簀倀倀✀簀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 㴀 ⠀砀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 砀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 ⬀ ⠀礀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 礀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 ⬀ ⠀稀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 稀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀㰀⼀栀㌀㸀ഀഀ ਍吀栀甀猀㨀㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀簀倀倀✀簀 㴀 ☀⌀㠀㜀㌀　㬀 ⠀ ⠀砀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 砀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 ⬀ ⠀礀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 礀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀⤀ ⬀ ⠀稀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 稀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀⤀ ⤀㰀⼀栀㌀㸀ഀഀ ਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀㸀ഀഀ

In Rⁿ:

਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍匀甀瀀瀀漀猀攀 眀攀 栀愀瘀攀 琀眀漀 瀀漀椀渀琀猀 ∀倀∀ 愀渀搀 ∀儀∀ 椀渀 刀㰀猀甀瀀㸀渀㰀⼀猀甀瀀㸀Ⰰ 眀椀琀栀 琀栀攀 挀漀漀爀搀椀渀愀琀攀猀㨀㰀戀爀㸀ഀഀ
਍⠀瀀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀Ⰰ 瀀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀Ⰰ ⸀⸀ Ⰰ瀀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀⤀ 愀渀搀 ⠀焀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀Ⰰ 焀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀Ⰰ ⸀⸀ Ⰰ焀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀⤀㰀戀爀㸀ഀഀ
਍琀栀攀渀㨀㰀戀爀㸀ഀഀ
਍㰀栀㌀㸀簀倀儀簀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 㴀 ⠀瀀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀 ⴀ 焀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 ⬀ ⸀⸀ ⬀ ⠀瀀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀 ⴀ 焀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀⤀㰀⼀栀㌀㸀ഀഀ ਍吀栀甀猀㨀㰀戀爀㸀ഀഀ ਍㰀栀㌀㸀簀倀儀簀 㴀 ☀⌀㠀㜀㌀　㬀 ⠀ ⠀瀀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀 ⴀ 焀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 ⬀ ⸀⸀ ⬀ ⠀瀀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀 ⴀ 焀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀⤀ ⤀㰀⼀栀㌀㸀ഀഀ ਍䤀渀 最攀渀攀爀愀氀Ⰰ 愀 ∀搀椀猀琀愀渀挀攀∀Ⰰ 椀猀 焀甀椀琀攀 漀昀琀攀渀 搀攀渀漀琀攀搀 戀礀 ∀匀∀ 漀爀 ∀猀∀Ⰰ 猀漀 礀漀甀 洀愀礀 愀氀猀漀 甀猀攀 琀栀愀琀 椀渀 攀焀甀愀琀椀漀渀猀⸀㰀戀爀㸀ഀഀ
਍一漀琀攀 琀栀愀琀 愀 琀攀爀洀 氀椀欀攀 ⠀瀀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀 ⴀ 焀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀 椀猀 攀焀甀椀瘀愀氀攀渀琀 琀漀 ⠀焀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀 ⴀ 瀀㰀猀甀戀㸀渀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀⸀㰀戀爀㸀ഀഀ This is so since the two have the same absolute value, but may differ in "sign" (+ or -). But, since we square it, the end result is the same.
਍ഀഀ
਍㰀戀爀㸀ഀഀ ਍㰀栀㄀㸀䌀栀愀瀀琀攀爀 㔀⸀ 䤀渀琀攀爀猀攀挀琀椀漀渀猀 漀昀 氀椀渀攀猀㨀㰀⼀栀㄀㸀ഀഀ ਍ഀഀ ਍㰀栀㌀㸀䔀砀愀洀瀀氀攀 漀昀 琀栀攀 ∀挀愀氀挀甀氀甀猀∀ 洀攀琀栀漀搀 椀渀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀㨀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ In note 2 (linear equations), we investigated how to find the "intersection" of two lines in R².
਍䤀渀 刀㰀猀甀瀀㸀㈀㰀⼀猀甀瀀㸀Ⰰ 椀昀 琀眀漀 氀椀渀攀猀 愀爀攀 渀漀琀 ∀瀀愀爀愀氀氀攀氀∀Ⰰ 琀栀攀渀 琀栀攀礀 眀椀氀氀 椀渀琀攀爀猀攀挀琀 ∀猀漀洀攀眀栀攀爀攀∀⸀㰀戀爀㸀ഀഀ
਍䤀渀 渀漀琀攀 ㈀Ⰰ 眀攀 甀猀攀搀 琀栀攀 昀漀氀氀漀眀椀渀最 攀砀愀洀瀀氀攀㨀㰀戀爀㸀ഀഀ
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀㸀ഀഀ Suppose we have the following two lines:
਍㰀戀爀㸀ഀഀ y= -3x - 3
਍礀㴀 ⴀ砀 ⴀ ㄀㰀戀爀㸀ഀഀ
਍䄀琀 眀栀椀挀栀 瀀漀椀渀琀Ⰰ 搀漀 琀栀攀礀 椀渀琀攀爀猀攀挀琀㼀㰀戀爀㸀ഀഀ
਍伀渀攀 琀栀椀渀最 椀猀 昀漀爀 猀甀爀攀㨀 琀栀攀 瀀漀椀渀琀 眀栀攀爀攀 琀栀攀 氀椀渀攀猀 椀渀琀攀爀猀攀挀琀Ⰰ 椀猀 琀栀攀 猀愀洀攀 ⠀砀Ⰰ礀⤀ 昀漀爀 戀漀琀栀 攀焀甀愀琀椀漀渀猀⸀㰀戀爀㸀ഀഀ Thus we may say:
਍㰀戀爀㸀ഀഀ -3x + 3 = -x - 1   =>
਍ⴀ㈀砀 㴀 ⴀ㐀 ☀渀戀猀瀀㬀 㴀㸀㰀戀爀㸀ഀഀ x = 2.
਍㰀戀爀㸀ഀഀ Thus at the intersection, x must be "2". Now use one of both equations to find the corresponding y. Let's use "y= -x - 1" :
਍㰀戀爀㸀ഀഀ y = -x -1 => y = -2 -1 => y = -3.
਍㰀戀爀㸀ഀഀ So, the point where both lines intersect is (2,-3).
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀甀攀∀㸀ഀഀ

Example of the "vector" method in R³:

਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀愀挀欀∀㸀ഀഀ ਍匀椀渀挀攀 琀栀攀 搀愀眀渀 漀昀 洀愀渀欀椀渀搀Ⰰ 漀爀 猀氀椀最栀琀氀礀 氀愀琀攀爀Ⰰ 椀琀 眀愀猀 愀氀爀攀愀搀礀 欀渀漀眀渀 琀栀愀琀 礀漀甀 漀渀氀礀 渀攀攀搀 ㈀ 瀀漀椀渀琀猀 琀漀 挀漀洀瀀氀攀琀攀氀礀 搀攀昀椀渀攀 愀 氀椀渀攀⸀㰀戀爀㸀ഀഀ
਍䄀最愀椀渀 眀攀 愀爀攀 最漀椀渀最 琀漀 猀攀攀 栀漀眀 琀漀 搀攀琀攀爀洀椀渀攀 眀栀攀爀攀 ⠀愀琀 眀栀椀挀栀 瀀漀椀渀琀⤀ 琀眀漀 氀椀渀攀猀 椀渀琀攀爀猀攀挀琀⸀ 䠀漀眀攀瘀攀爀Ⰰ 琀栀椀猀 琀椀洀攀 眀攀 眀椀氀氀 甀猀攀 氀椀渀攀愀爀 愀氀最攀戀爀愀⸀㰀戀爀㸀ഀഀ The method shown here, works the same way in vectorspaces of all dimensions (e.g. R³, R⁴ etc...).
਍㰀戀爀㸀ഀഀ In figure 5 below, I tried to draw 2 lines in a Cartesian coordinate system (R³).
਍䠀漀瀀攀昀甀氀氀礀Ⰰ 礀漀甀 挀愀渀 猀攀攀 愀 最爀攀攀渀 氀椀渀攀 愀渀搀 愀渀 漀爀愀渀最攀 漀渀攀⸀㰀戀爀㸀ഀഀ
਍ഀഀ ਍ഀഀ Figure 5. Example lines in R³ (2 points A and B is enough to describe any line)
਍㰀戀爀㸀ഀഀ ਍㰀戀爀㸀ഀഀ
਍ⴀ䰀攀琀✀猀 昀漀挀甀猀 漀渀 琀栀攀 最爀攀攀渀 氀椀渀攀 昀漀爀 愀 洀漀洀攀渀琀⸀㰀戀爀㸀ഀഀ
਍匀愀礀 琀栀愀琀 眀攀 栀愀瘀攀 琀眀漀 瀀漀椀渀琀猀 漀渀 琀栀愀琀 氀椀渀攀Ⰰ 䄀 愀渀搀 䈀⸀ 吀栀攀渀 眀攀 愀氀猀漀 栀愀瘀攀 琀栀攀 瘀攀挀琀漀爀猀 㰀䈀㸀䄀㰀⼀䈀㸀 愀渀搀 㰀䈀㸀䈀㰀⼀䈀㸀Ⰰ㰀戀爀㸀ഀഀ depicted as arrows, originating from the origin.
਍䈀礀 琀栀攀 眀愀礀Ⰰ 椀琀 搀漀攀猀 渀漀琀 洀愀琀琀攀爀 愀琀 愀氀氀Ⰰ 眀栀攀爀攀 琀栀漀猀攀 琀眀漀 瀀漀椀渀琀猀 愀爀攀Ⰰ 愀猀 氀漀渀最 愀猀 琀栀攀礀 愀爀攀 漀渀 琀栀愀琀 氀椀渀攀⸀㰀戀爀㸀ഀഀ
਍夀漀甀 洀愀礀 眀漀渀搀攀爀 栀漀眀 眀攀 挀愀渀 眀爀椀琀攀 搀漀眀渀 愀渀 攀焀甀愀琀椀漀渀Ⰰ 戀礀 眀栀椀挀栀 眀攀 挀愀渀 ∀愀搀搀爀攀猀猀∀ 愀渀礀 爀愀渀搀漀洀 瀀漀椀渀琀 ⠀砀Ⰰ礀Ⰰ稀⤀ 漀渀 琀栀愀琀 氀椀渀攀⸀㰀戀爀㸀ഀഀ Well, here it is:
਍㰀戀爀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ x ┐
਍☀⌀㤀㐀㜀㐀㬀 礀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ z ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ A + λ (B - A) ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍䤀琀✀猀 樀甀猀琀 愀 洀愀琀琀攀爀 漀昀 瘀攀挀琀漀爀 愀搀搀椀琀椀漀渀⸀ 吀栀攀 瘀攀挀琀漀爀 ⠀㰀䈀㸀䈀㰀⼀䈀㸀 ⴀ 㰀䈀㸀䄀㰀⼀䈀㸀⤀Ⰰ 椀猀 琀栀攀 礀攀氀氀漀眀 瀀愀爀琀Ⰰ 愀渀搀 㰀䈀㸀氀椀攀猀 漀渀㰀⼀䈀㸀 琀栀攀 氀椀渀攀⸀㰀戀爀㸀ഀഀ By varying the scalar λ, you can reach any point on the line (e.g. λ=0.3, or λ=5 etc...).
਍㰀戀爀㸀ഀഀ You should see it this way: use vector A, to "step" on the line, and then you can use λ(B - A) to reach any point on the line.
਍㰀戀爀㸀ഀഀ -Now, focus on the orange line.
਍㰀戀爀㸀ഀഀ A similar argument can be used, and we can "address" any random point (x,y,z) on the orange line line, by using:
਍㰀戀爀㸀ഀഀ

਍㴀ഀഀ

਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ ┌ x ┐
਍☀⌀㤀㐀㜀㐀㬀 礀 ☀⌀㤀㐀㜀㐀㬀㰀戀爀㸀ഀഀ └ z ┘ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ C + μ (D - C) ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䄀䈀䰀䔀㸀ഀഀ
਍一漀眀Ⰰ 猀甀瀀瀀漀猀攀 椀渀 爀漀眀 瘀攀挀琀漀爀 渀漀琀愀琀椀漀渀Ⰰ 眀攀 栀愀瘀攀㨀㰀戀爀㸀ഀഀ
਍䄀 㴀 ⠀愀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀Ⰰ愀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀Ⰰ愀㰀猀甀戀㸀㌀㰀⼀猀甀戀㸀⤀㰀戀爀㸀ഀഀ B = (b₁,b₂,b₃)
਍䌀 㴀 ⠀挀㰀猀甀戀㸀㄀㰀⼀猀甀戀㸀Ⰰ挀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀Ⰰ挀㰀猀甀戀㸀㌀㰀⼀猀甀戀㸀⤀㰀戀爀㸀ഀഀ D = (d₁,d₂,d₃)
਍㰀戀爀㸀ഀഀ If the two lines really intersect, then at that point, we have the same x, y, z.
਍㰀戀爀㸀ഀഀ Thus, it must hold at that point, that:
਍㰀戀爀㸀ഀഀ

਍㴀ഀഀ

਍㰀吀䐀㸀 㰀昀漀渀琀 挀漀氀漀爀㴀∀戀爀漀眀渀∀ 昀愀挀攀㴀∀挀漀甀爀椀攀爀∀㸀ഀഀ A + λ (B - A) ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀䐀㸀ഀഀ ਍㰀⼀吀刀㸀ഀഀ

਍㴀ഀഀ	਍㰀䈀㸀䌀㰀⼀䈀㸀 ⬀ ☀⌀㤀㔀㘀㬀 ⠀㰀䈀㸀䐀㰀⼀䈀㸀 ⴀ 㰀䈀㸀䌀㰀⼀䈀㸀⤀ഀഀ

਍㰀戀爀㸀ഀഀ If we write that out, we have a system of 3 linear equations:
਍㰀戀爀㸀ഀഀ a₁ + λ (b₁ - a₁) = c₁ + μ (d₁ - c₁)
਍愀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⬀ ☀⌀㤀㔀㔀㬀 ⠀戀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 愀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀⤀ 㴀 挀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⬀ ☀⌀㤀㔀㘀㬀 ⠀搀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀 ⴀ 挀㰀猀甀戀㸀㈀㰀⼀猀甀戀㸀⤀㰀戀爀㸀ഀഀ a₃ + λ (b₃ - a₃) = c₃ + μ (d₃ - c₃)
਍㰀戀爀㸀ഀഀ Do not forget that the a₁, b₁ etc..., are simply known values, since we know the coordinates of A, B, C, and D.
਍吀栀攀 猀攀琀 漀昀 攀焀甀愀琀椀漀渀猀 琀栀攀渀 戀漀椀氀猀 搀漀眀渀 琀漀 爀攀猀漀氀瘀椀渀最 ☀⌀㤀㔀㔀㬀 愀渀搀 ☀⌀㤀㔀㘀㬀Ⰰ 眀栀椀挀栀 椀猀 瀀漀猀猀椀戀氀攀 戀礀 猀甀戀猀琀椀琀甀琀椀渀最⸀㰀戀爀㸀ഀഀ I am not going to work this out any further. For me, it's only important that you grasp the method of handling this problem.
਍㰀戀爀㸀ഀഀ I agree that solving the set of three linear equations is quite some work, but the method is generic for multiple dimensions.
਍匀甀瀀瀀漀猀攀 眀攀 栀愀瘀攀 琀栀攀 猀愀洀攀 瀀爀漀戀氀攀洀 椀渀 刀㰀猀甀瀀㸀㐀㰀⼀猀甀瀀㸀⸀ 吀栀攀渀 漀渀氀礀 漀渀攀 攀砀琀爀愀 挀漀漀爀搀椀渀愀琀攀 椀猀 椀渀瘀漀氀瘀攀搀 ⠀昀漀爀 愀氀氀 㐀 瀀漀椀渀琀猀⤀Ⰰ㰀戀爀㸀ഀഀ but the principle really stays the same.
਍ഀഀ ਍ഀഀ
਍㰀戀爀㸀ഀഀ
਍㰀昀漀渀琀 挀漀氀漀爀㴀∀戀氀甀攀∀㸀ഀഀ

That's it. Hope you liked it !

਍㰀栀㌀㸀㰀䤀㸀吀栀攀 渀攀砀琀 渀漀琀攀 眀椀氀氀 攀砀瀀氀漀爀攀 猀漀洀攀 洀漀爀攀 愀搀瘀愀渀挀攀搀 搀椀昀昀攀爀攀渀琀椀愀氀猀 氀椀欀攀 琀栀攀 䜀爀愀搀Ⰰ 䐀椀昀昀Ⰰ 愀渀搀 䌀甀爀氀 漀瀀攀爀愀琀漀爀猀⸀㰀⼀䤀㸀㰀⼀栀㌀㸀ഀഀ ਍ഀഀ
਍㰀戀爀㸀ഀഀ
਍㰀戀爀㸀ഀഀ
਍㰀⼀戀漀搀礀㸀ഀഀ