Question

I only have a few hours to answer and send. I would appreciate if you help. Thank you so much.

a 1. Let V = {[ :a+c=b+ =b+ d} une and T:V + R with T ([: 1) = a +c. Cd a) Find a basis for the kernel of T. dim(Ker(T)) =? (10P) b) Find a basis for the image of T. dim(Im(T)) =? (10P) c) Is T an isomorphism? (5P) 2. Let T = {(2,3), (3, 2)} be a basis for R2 and S = {(1,0,0), (0,1,0),(0,0,1)} be the standard basis for R3. Determine the linear transformation L : R3 + R2 such that the matrix T[L]s of L relative to the -1 0-1 bases S and T is A 0 1 1] 1 3. Consider the basis S = {(1,0, -1),(-1,1,0), (0,1,1)} of R3. Apply Gram-Schmidt orthogonalization process to S and find an orthonormal basis for R3. 4. Find the eigenvalues and the eigenvectors corresponding to the eigenvalues of the matrix A -4 0 -6 0 -1 0 3 0 5

Answer 1

T: VR atc dim v= 4- No. of L.I restriction 4-1 - 3 dimv=3 kert {C}ev / "[24] = 0? - atc= {[EN] í c=-a & ate=bra a b cd bt d=0 [ C+Q=0 a b d=-b -a-b C=-a Idim kert S ree By Rank Nullity Thim dimv=Y (7) + dim(kert) dim (Int) +2 3

dim (IMT) = 3-2 =) dim Imt =) و Fer isomorphism dim uz dim R 3 + 1 So, I us not au Isomarplusm T.