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**1) Explain what can be said about det A (determinant of A) if:**

A^2 + I = 0, A is n x n

A^2 + I = 0, A is n x n

My attempt:

A^2 = -I

(det A)^2 = (-1)^n

If n is be even, then det A = 1 or -1

But what happens when n is ODD, what is the detmerinant of A, then?

**2) If V is a subspace of R^n and W is a subspace of V, then W^**

__|__is a subspace of V^__|__. True or False? Justify. [Note:__|__means orthogonal complement][By dimension theorem, dim dim V + dim V^

__|__= n, but is it true that dim W + dim W^

__|__= n also?

If V is a subspace of R^n and W is a subspace of V, does this IMPLY that W is also a subspace of R^n? WHY or why not?

I understand that if A is a subset of B and B is a subset of C, then A is a subset of C, but subspace and subset are not the same thing, so I am not too sure...]

**3) Let T: R^3->R^3 be a linear transformation.**

ker (T) = span {[1 1 1]^T, [1 -2 1]^T}, and T([3 2 1]^T)=[10 10 10]^T. Find the matrix of T.

ker (T) = span {[1 1 1]^T, [1 -2 1]^T}, and T([3 2 1]^T)=[10 10 10]^T. Find the matrix of T.

I have no idea how to do this one...

Thanks for your help!