Hi Everyone, I firstly want to apologize for the long post, but I have a Linear Algebra Exam on Wednesday, and We received a review sheet in class today, but no answers. I've been working on it for a while and would really appreciate some affirmation, and clearing up.

1) Let v be a non zero vector in Rn, and let W={x in Rn such that x^{T}*v=0} Show that W is a subspace of dimension (n-1)

I can show that W is in fact a subspace through the closure properties, however I'm having trouble finding a minimal spanning set with n-1 vectors in it.

2) Let L: Rn to Rm be a linear transformation, and W a subspace of Rm. Let L_{W} = {x in Rn given L(x) is in W} Show that L_{W }is a subspace of Rn.

Here I believe it is just showing the zero vector is contained and that the 2 closure properties follow, but I'm not sure about showing this with a Linear Transformation.

3) Let P be an n x n matrix satisfying the equation P^{m. }= P for some m > 1 Suppose that N(P) = {0}. Show that if P is diagonalizable, then P^{2} = Id (Identity Matrix)

This question really has me puzzled right now.

4) Suppose T: Rm to Rn is a linear transformation, and {v_{1},.....,v_{k} } are k vectors in Rm for which {T(v_{1}),.....,T(v_{k})} are linearly independent in Rn. Show that {v_{1},.....,v_{k}} are linearly independent in Rm.

This makes sense to me that any vectors having undergone a linear transformation, and are still linearly independent, must have been linearly independent beforehand, but I'm lacking on a more mathematically driven way to show it.

Lastly True False, with explanations regarding why it is true or false. I haven't attempted these yet so have no current answers, but am working on them tonight and hoping this can be answered by tomorrow night so I may use it to study.

A) If A and B are n x n matrices, then Det(A+B) = Det(A) + Det(B).

B) If A is nonsingular and diagonalizable, then A^{-1}is also diagonalizable

C) If S = {v_{1},v_{2},.....,v_{n}} is a basis for Rn and W is a subspace of Rn, then SnRn = {v_{1} given v_{1}is in W} is a basis for W.

D) Let W C Rn be a subspace, and let W(Perpendicular sign) = {y in Rn such that x (dot) y =0 for all x in W}. Then W(perpendicular) is a subspace of Rn

If anyone could reply with what the work should look like, or an explanation of the steps for these problems, I would greatly appreciate it so that I can check my work and be more prepared for Wednesday.

Thanks Everyone!