Description
1. Let e1 =
Å
1
0
ã
, e2 =
Å
0
1
ã
, V = span(e1, e2 − e1), W = span(e2) be two
subspaces of C
2
, prove that C
2 = V + W but it is not a direct sum.
2. Let V = span{e1, e2}, W = span{e2, e3}, where e1, e2, e3 are vectors in R
3
,
prove R
3 = V + W. Is the sum a direct sum? Justify your answer.
3. Let A be a m × n complex valued matrix, use SVD to prove that C
m =
range(A) ⊕ null(A∗
).
