Linear Algebra, 2020a

Section VI. Projection 287
We will compare the projections of this element of R 3
⎛ ⎞
2
⎜ ⎟
⃗v = ⎝2⎠
5
into M along these two subspaces (verification that R 3 = M⊕N and R 3 = M⊕ ˆN
is routine).
⎛
⎜
0
⎞
⎛ ⎞
0
⎟
⎜ ⎟
N = {k ⎝0⎠ | k ∈ R} ˆN = {k ⎝ 1 ⎠ | k ∈ R}
1
−2
Here are natural bases for N and ˆN.
⎛
⎜
0
⎞
⎛ ⎞
0
⎟
B N = 〈 ⎝0⎠〉 B ˆN = 〈 ⎜ ⎟
⎝ 1 ⎠〉
1
−2
⌢
To project into M along N, represent ⃗v with respect to the concatenation B M BN
⎛
⎜
2
⎞ ⎛
⎟ ⎜
1
⎞ ⎛
⎟ ⎜
0
⎞ ⎛
⎟ ⎜
0
⎞
⎟
⎝2⎠ = 2 · ⎝0⎠ + 1 · ⎝2⎠ + 4 · ⎝0⎠
5 0 1 1
and drop the N term.
⎛
⎜
1
⎞ ⎛
⎟ ⎜
0
⎞ ⎛
⎟ ⎜
2
⎞
⎟
proj M,N (⃗v )=2 · ⎝0⎠ + 1 · ⎝2⎠ = ⎝2⎠
0 1 1
⌢
To project into M along ˆN represent ⃗v with respect to B M B ˆN
⎛ ⎞ ⎛ ⎞ ⎛
2 1
⎜ ⎟ ⎜ ⎟ ⎜
0
⎞ ⎛ ⎞
0
⎟ ⎜ ⎟
⎝2⎠ = 2 · ⎝0⎠ +(9/5) · ⎝2⎠ −(8/5) · ⎝ 1 ⎠
5 0
1
−2
and omit the ˆN part.
⎛
proj M, ˆN (⃗v )=2 · ⎜
1
⎞ ⎛
⎟ ⎜
0
⎞ ⎛ ⎞
2
⎟ ⎜ ⎟
⎝0⎠ +(9/5) · ⎝2⎠ = ⎝18/5⎠
0
1 9/5
So projecting along different subspaces can give different results.
These pictures compare the two maps. Both show that the projection is
indeed ‘into’ the plane and ‘along’ the line.