You are given that v is non zero so at least one component of v is non-zero. Let vi be a non-zero component. Define u1= <1, 0, ..., ai, 0, ..., 0> such v1+ aivi= 0. That is, the first component is 1, the ith component is v1/vi, and all other components are 0. Define u2= <0, 1, ..., bi, 0, ..., 0> such that v2+ bivi= 0 and all other components are 0. That is, the second component is 1, the ith component is v2/vi and all other components are 0. Continue in that way defining uj to be the vector such the jth component is 1 for all j= 1 to n but j NOT equal to I, the ith component vj/vi, and all other components are 0. That will give n- 1 vectors that, by construction, have dot product with v equal to 0. It needs to be shown that they are independent and so span W.

Yes, and it is pretty straight forward. If u and v are in L2) 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._{W}, the L(u) is in W and L(v) is in W. All you need to show is that L(u+ v)= L(u)+ L(v) is in W and L(ku)= kL(u), where k is a scalar, is in W.

Prove it by contradiction. If they are NOT linearly independent, then there exist scalars, a1, a2, ..., ak, not all 0, such that a1v1+ a2v2+ ...+ akvk= 0. Then T(a1v1+ a2v2+ ...+ akvk)= a1T(v1)+ a2T(v2)+ ...+ akT(vk)= T(0)= 0.3) 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.

[quote]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).[/tex]

Look at and .

A is diagonalizable if and only if there exist a matrix B such that where D is the diagonal matrix having the eigenvalues of A on its main diagonal. Since A is nonsingular, none of those is 0. Show that and is the diagonal matrix having theB) If A is nonsingular and diagonalizable, then A^{-1}is also diagonalizablereciprocalsof the eigenvalues of A on its diagonal.

Are you sure you have written this correctly? What you have written is that, given any basis for Rn and any subspace, W, of Rn, theC) 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.firstvector in the basis of Rn is a basis for W. Although it is NOT what you have written, if you replace v_{1}with v_{i}, you would be saying that selecting a subset from the first basis, of all basis vectors that are in W, you have a basis for W. That sounds better but is still NOT true. For example, {<1, 0>, <0, 1>} is a basis for R2 and W= {<x, y>| x+ y= 0} is a subspace of R2 but neither vector in {< 1, 0>, <0, 1> } is in W.

Again, it is just a matter of checking that the sum of two vectors in the set is in the set and that the scalar product of a vector in the set with any scalar is in the set.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

Suppose u and v are in W. Then x.u= 0 and x.v= 0 for all x in W. So what is x.(u+ v)? What is x.(ku) for any k?

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!