Question 16. [3 W of R\" from L19.4 (or textbook, Theorem 7.11): If A is a matri
ID: 3116512 • Letter: Q
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Question 16. [3 W of R" from L19.4 (or textbook, Theorem 7.11): If A is a matrix whose columns form a basis of W, then projW(2) = A (ATA)-1 Ax for all vectors x in R". Now let points Recall the formula for the projection onto a subspace Calculate the projection matrix B-A(ATA)-1A. (Hint. There are two ways to do this. Either think about what the subspace W, which is the span of the columns of A, actually is, OR multiply out the various matrices. If you want to multiply the matrices, first work out A1 A, then its inverse (A' A) and then B.)Explanation / Answer
We have col(A) = span{(2,0,0,0)T,(0,1,0,0)T,(0,0,1,0)T}. Let u1 =(1,0,0,0)T, u2 =(0,1,0,0)T and u3 =(0,0,1,0)T We can now compute proju1(x) = [(x.u1)/(u1.u1)]u1. Similarly, proju2(x) and proju3(x) can also be computed. Then projW(x) = proju1(x)+ proju2(x)+ proju3(x).
The other method is to compute the projection matrix A(ATA)-1 A . Now A is a 4x3 matrix so that AT is a 3x4 matrix. Then ATA ( a 3x3 matrix) =
4
0
0
0
1
0
0
0
1
Also, (ATA)-1 =
1/4
0
-1/4
0
1
0
0
0
1
i.e. another 3x3 matrix. On multipplying this 3x3 matrix to the left by A (a 4x3 matrix), we will get a 4x3 matrix A(ATA)-1 =
1/2
0
-1/2
0
1
0
0
0
1
0
0
0
which is a 4x3 matrix. Now, we cannot multiply this 4x3 matrix to the right by A which is another4x3 matrix. Thus, the matrix A(ATA)-1 A does not exist.
Note: Without actually computing (ATA), (ATA)-1, A(ATA)-1 , it is known that A(ATA)-1 A does not exist as matrix multiplication is row by column.
4
0
0
0
1
0
0
0
1
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