Recent content by nautolian

Oh, I see. Thank you. So for the Range, because they give you what the system of equations equals, would that be your b? or would you solve for Ax=b where b is, say, (x,y,z)? Thanks again.

Homework Statement Consider the left multiplication L_A:R^4->R^3 corresponding the matrix A: 1 2 -1 3 = 2 2 4 -1 6 = 5 0 1 0 2 = 3 What is the rank of L_A and the Range of L_a? Homework Equations The Attempt at a Solution I have two main problems with this question...

I thought it might be an eigenvector because i was under the impression for some reason that the orthonormalization would no necessarily mean using the Gram-Shmidt process, but rather just normalizing it (dividing by sqrt(sum of squares of eigenvector values)). Is this not the case?

Why aren't those eigenvectors? Because you've changed the vectors too much?

Homework Statement in light of the gram-schmidt orthogonalization process, if A: Rn -> Rn and we have a basis of Rn of eigenvectors of A, can't we just orthonormalize them and get a matrix P such that P-1=PT and thus PTAP is diagonal? Homework Equations The Attempt at a Solution I...

okay, would that mean that there are infinite eigenvalues with associated iegenvectors equal to A*exp(lambda*x)? where A is a constant? Sorry I'm still unclear about this. Thanks for the help though!

High, sorry I'm still not really sure where to go with this. I mean I understand that Df=(lambda)f, but in the terms of C^1(R) does this mean that the derivative of the complex number a+bi is the same as the eigenvalue times the vector? Sorry, I'm still somewhat lost. Thanks for your help though.

Homework Statement Find the eigenvectors and eigenvalues of the differentiation map C1(R) -> C1(R) from the vector space of differentiable functions to itself. Homework Equations The Attempt at a Solution Hi, I'm not entirely sure how to go about this, because would the...

Homework Statement Show that if ker T != 0 then T is not an isomorphism. Homework Equations The Attempt at a Solution If Ker T != 0 that means that there are multiple solutions for which T=0 meaning it is not injective and hence not isomorphic? Is that correct? I don't think it...

Homework Statement Let V be an inner product space. For v ∈ V fixed, show that T(u) =< v, u > is a linear operator on V . Homework Equations The Attempt at a Solution First to show it is a linear operator, you show that T(u+g)=T(u)+T(g) and T(ku)=kT(u) So, T(u+g)=<v...

Could you do a combination say (0 a, 0 -b) (0 -b, 0 a), (a 0, b 0), (-b 0, a 0)? Could this form a basis? Or does it have to be real integers? thanks

Homework Statement Show that B = {x2 −1,2x2 +x−3,3x2 +x} is a basis for P2(R). Show that the differentiation map D : P2(R) → P2(R) is a linear transformation. Finally, find the following matrix representations of D: DSt←St, DSt←B and DB←B. Homework Equations The Attempt at a...

Okay, so would I say kernel is {A where tr(A)=0, for A in Mnn}? So what would the image be then? Thanks

Homework Statement Let F : Mnn(R) → R where F(A) =tr(A). Show that F is a linear transformation. Find the kernel of F as well as its dimension. What is the image of F? Homework Equations The Attempt at a Solution I have shown that it is a linear transformation. But I am not...

Homework Statement The set K of 2 × 2 real matrices of the form [a b, -b a] form a field with the usual operations. It should be clear to you that M22(R) is a vector space over K. What is the dimension of M22(R) over K? Justify your answer by displaying a basis and proving that the set...