Indicate whether each statement is true or false True False

Indicate whether each statement is true or false.

[True | False] Define two subspaces V and W of R^3 by V = Span(v_1, v_2) and W = Span(w_1, w_2), where v_1 = (1, 3, 2), v_2 = (0, 15, 10). w_1 = (1, 0, 0), w_2 = (0, -3, -2). Then V and W are different subspaces of R^3. [True | False] If A is an n times n matrix whose row space and column space are equal, as subspaces of R^n, then A is a symmetric matrix. [True | False] If A is an n times n matrix which is diagonalizable, then A has n distinct eigenvalues. [True | False] If I: R^n rightarrow R^n is the identity map, and X and Y are any two bases of R^n. then the matrix Y Ix (of I with respect to X and Y) is the identity matrix. [True | False] If 0 is the only eigenvalue of a square matrix A, then A is the zero matrix.

Solution

1

0

1

0

0

1

-1/5

-1/5

0

0

0

0

This indicates that w₁ = v₁-v₂/5 and w₂ = -v₂/5 . Thus V and W are same.

2. The statement is True. If A is a nxn matrix whose row and column spaces are same as subspaces of Rⁿ, then A and A^T have the same column space so that the RREFs of A and A^T are identical. This means that A is a symmetric matrix.

3.The statement is False. If a nxn matrix A is diagonalizable, then it has n distinct and linearly independent eigenvectors. Some eigenvalues can be of multiplicity more than 1.

4.The statement is True. I, the identity matrix, represents the identity map, regardless to any ordered bases.

5. The statement is False. If 0 is the only eigenvalue of a nxn matrix A, then 0 is the only solution to the equation det(A- I_n) =0 ,then det(A) = 0, then A is not invertible. A need not be 0 matrix.

1	0	1	0
0	1	-1/5	-1/5
0	0	0	0