Recent content by specialnlovin

True or false. provide either a proof or counter example accordingly if f is a function V\timesV\rightarrowk such that for all v,u,w\inV, \lambda\ink, f(\lambdav+u,w)=\lambdaf(v,w)+f(u,w). Then f is bilinear I know that this does not include the second part of the requirement to be bilinear...

the largest jordan block of 1 is 3x3 and 2 is 1x1. so the eigenspace of 1 is the span ((2, 3, 0, 1)T, (-1, -1, 1, 0)T, (0, 1, 0, 0)T) and the eigenspace of 2 is the span of (1, 1, 1, 1)T right?

For V1(1) i found the vector v1=(2, 3, 0, 1)T for V2(1) i found the vector v2=(-1, -1, 1 0)T but I cannot figure out v3

(a-i)3(a-2i)

so is the eigenspace of 1 the span of (2, 3, 0, 1)t? and the eigenspace of 2 the span of (1, 1, 1, 1)t? but then how do i get the eigenvectors from that?

how do we find the basis of eigenvectors though? when i did (A-I) i got only one vector

so the largest sizes of the jordan blocks for 1 and 2 are 1x1 blocks

Let A=2 -1 2 1 1 0 3 1 -2 1 0 1 -1 0 0 3 the characteristic polynomial of A is (x-1)3(x-2) find the minimal polynomial, jordan basis, and jordan normal form I know the minimum polynomial is (x-1)(x-2), but I am not sure how to find the nordan basis and jordan normal form

All i can think of is f(x)=1 or f(T)=I

Let V =Mn(k),n>1 and T:V→V defined by T(M)=Mt (transpose of M). i) Find the minimal polynomial of T. Is T diagonalisable when k = R,C,F2? ii) Suppose k = R. Find the characteristic polynomial chT . I know that T2=T(Mt))=M and that has got to help me find the minimal polynomial

Let V=C[x]10 be the fector space of polynomials over C of degree less than 10 and let D:V\rightarrowV be the linear map defined by D(f)=f' where f' denotes the derivatige. Show that D11=0 and deduce that 0 is the only eigenvalue of D. find a basis for the generalized eigenspaces V1(0), V2(0)...

Let V=Mn(k) be a vector space of matrices with entries in k. For a matrix M denote the trace of M by tr(M). What is the dimension of the subspace of {M\inV: tr(M)=0} I know that I am supposed to use the rank-nullity theorem. However I'm not sure exactly how to use it. I know that the trace is...

right, that just seemed way too easy I thought I was doing it wrong. Then the im(T) is the span of e2, e3, and e4

Okay so matrix T(x)=x2 and with respect to the basis {1,x,x²,x³} the second column would be (0,0,1,0), T(x2)=x3 and with respect to the basis {1,x,x²,x³} would be (0,0,0,1). I would then say that T=(0 0 0) (1 0 0) (0 1 0) (0 0 1) with respect to {1,x,x²,x³} right? Not with respect...

Let T: R[x]2\rightarrow R[x]3 be defined by T(P(x))=xP(x). Compute the matrix of x with respect to bases {1,x,x2} and {1,x,x2,x3}. Find the kernel and image of T. I know how to do this when given bases without exponents, however I do not know exactly what this is saying and therefore am...