Linear Algebra, 2020a

286 Chapter Three. Maps Between Spaces
3.1 Definition Let a vector space be a direct sum V = M ⊕ N. Then for any
⃗v ∈ V with ⃗v = ⃗m + ⃗n where ⃗m ∈ M, ⃗n ∈ N, the projection of ⃗v into M along
N is proj M,N (⃗v )= ⃗m.
This definition applies in spaces where we don’t have a ready definition
of orthogonal. (Definitions of orthogonality for spaces other than the R n are
perfectly possible but we haven’t seen any in this book.)
3.2 Example The space M 2×2 of 2×2 matrices is the direct sum of these two.
( )
( )
a b
0 0
M = { | a, b ∈ R} N = { | c, d ∈ R}
0 0
c d
To project
( )
3 1
A =
0 4
into M along N, we first fix bases for the two subspaces.
( ) ( )
( ) ( )
1 0 0 1
0 0 0 0
B M = 〈 , 〉 B N = 〈 , 〉
0 0 0 0
1 0 0 1
Their concatenation
( ) ( ) ( ) ( )
⌢ 1 0 0 1 0 0 0 0
B = B M BN = 〈 , , , 〉
0 0 0 0 1 0 0 1
is a basis for the entire space because M 2×2 is the direct sum. So we can use it
to represent A.
( ) ( ) ( ) ( ) ( )
3 1 1 0 0 1 0 0 0 0
= 3 · + 1 · + 0 · + 4 ·
0 4 0 0 0 0 1 0 0 1
The projection of A into M along N keeps the M part and drops the N part.
( ) ( ) ( ) ( )
3 1 1 0 0 1 3 1
proj M,N ( )=3 · + 1 · =
0 4 0 0 0 0 0 0
3.3 Example Both subscripts on proj M,N (⃗v ) are significant. The first subscript
M matters because the result of the projection is a member of M. For an
example showing that the second one matters, fix this plane subspace of R 3 and
its basis.
⎛ ⎞
⎛ ⎞ ⎛
x
1
⎜ ⎟
⎜ ⎟ ⎜
0
⎞
⎟
M = { ⎝y⎠ | y − 2z = 0} B M = 〈 ⎝0⎠ , ⎝2⎠〉
z
0 1