1. Let V be a vector space over R and U; W V be two subspaces of V .
i. Prove that there exist a basis B of U and a basis C of W such that B \ C is a basis for U \ W .
Is it true that for every basis B of U and every basis C of W the set B \ C is a basis for U \ W ?
Recall the de nition of U + W from previous HW's. Prove the following dimension formula:
dim(U + W ) = dim(U) + dim(W ) dim(U \ W ):
(Important Remark: This dimension formula can be used in later HW's and in exams without adding a proof to it.)
iv. Let U; W R4[x] be two subspaces which satisfy dim (U) = dim(W ) = 3 prove that U \ W ≠ f0g.
v. Find the dimension of the following space:
1;
0
1
span
0
1
1; 0
2
1
span
0
1
1; 0
1
2
f
B
1
C B
1
C
g \
f
B
0
C B
2
C B
1
C
g
1
1
1
3
1
B
C B
C
B
C B
C B
C
@
2
A @
3
A
@
2
A @
2
A @
4
A
The following claims are either true or false. Determine which case is it for each claim and prove your answer.
There exists a matrix A 2 M4 3 who's last row is not a linear combina-tion of the rest of it's rows.
There exists a matrix A 2 M3(R) which satis es rank(A) = 2 and
1
1
A@ 1 A=0:
1
iii. There exists a matrix A 2 M3(R) which satis es rank(A) = 2, and
0
1
1
0 11
1
A
1
= 0
A
= 0
@
1
A
@
1
A
.
There exist two matrices A; B 2 M3(R) such that A ≠ B, A s B (A is row equivalent to B) and the column spaces of A and B are the same.
There exist two matrices A; B 2 M3(R) such that AB = I and
rank(A) + rank(B) = 5:
1
2
There exists a matrix A 2 M3(R) who's row space is equal to its column space.
The following claims are either true or false. Determine which case is it for each claim and prove your answer.
For any two matrices A 2 Mm n(R) and B 2 Mn k(R) we have rank(AB) = rank(A) rank(B).
If A is a square matrix then its column space is equal to its null space.
The system (Ajb) has a solution i rankA=rank(Ajb)
If A 2 Mm n(R) is such that the vectors in its rows are linearly inde-pendent then the vectors in its columns are also linearly independent.
If A 2 Mn(R) is such that the vectors in its rows are linearly independent then the vectors in its columns are also linearly independent
Let A 2 Mn(R) prove that A is invertible i rank(A) = n.
For any two matrices A 2 Mm n(R) and B 2 Mn k(R) which satisfy AB = 0 prove that rank(B) + rank(A) n.
The following claims are either true or false. Determine which case is it for each claim and prove your answer.
For any two m n matrices A and B we have rank(A + B) = rank(A) + rank(B).
For any two m n matrices A and B we have rank(A + B) rank(A) + rank(B).
Let A 2 Mm n(R) and B 2 Mn k(R), consider the matrix AB and prove the following:
The null space of B is a subspace of the null space of AB.
The column space of AB is a subspace of the column space of A.
rank(AB) minfrank(A); rank(B)g. (Hint: use the previous parts of this question).
Let A 2 M3(R) be such that rank(A), rank(A2) and rank(A3) are three di erent numbers. Prove that A3 = 0.
