Linear Algebra

258 Chapter Three. Maps Between SpacesThe prior subsections project a vector into a line by decomposing it into twoparts: the part in the line proj [⃗s ] (⃗v ) and the rest ⃗v − proj [⃗s ] (⃗v ). To generalizeprojection to arbitrary subspaces we will follow this decomposition idea.3.1 Definition For any direct sum V = M ⊕ N and any ⃗v ∈ V, the projection of⃗v into M along N isproj M,N (⃗v ) = ⃗mwhere ⃗v = ⃗m + ⃗n with ⃗m ∈ M, ⃗n ∈ N.We can apply this definition in spaces where we don’t have a ready definitionof orthogonal. (Definitions of orthogonality for spaces other than the R n areperfectly 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 0c dTo project( )3 1A =0 4into M along N, we first fix bases for the two subspaces.( ) ( )( )1 0 0 10 0B M = 〈 , 〉 B N = 〈 ,0 0 0 01 0()0 0〉0 1The concatenation of these( ) ( ) ( ) ( )⌢ 1 0 0 1 0 0 0 0B = B M BN = 〈 , , , 〉0 0 0 0 1 0 0 1is a basis for the entire space because M 2×2 is the direct sum. So we can use itto 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 1The projection of A into M along N keeps the M part and drops the N part.( ) ( ) ( ) ( )3 1 1 0 0 1 3 1proj M,N ( ) = 3 · + 1 · =0 4 0 0 0 0 0 03.3 Example Both subscripts on proj M,N (⃗v ) are significant. The first subscriptM matters because the result of the projection is a member of M. For anexample showing that the second one matters, fix this plane subspace of R 3 andits basis.⎛ ⎞⎛ ⎞ ⎛x⎜ ⎟M = { ⎝y⎠ ∣ 1⎜ ⎟ ⎜0⎞⎟y − 2z = 0} BM = 〈 ⎝0⎠ , ⎝2⎠〉z0 1