Find the matrix A of T with respect to the standard basis...of both V and W. Compute the kernel, the image, the nullity and the rank of T.
(See attachment for full question)

I have two questions that I need help with.
1) How would you find a basis of the kernel, a basis of the image and determine the dimension of each for this matrix? The
matrix is in the attachment.
2) Are the following 3 vectors linearly dependent? (see attachment for the three vectors) How can you decide?
I hope y

Here's my problem:
If A and B are subsets of a group G, define
AB = {ab | a 2 A, b 2 B}. Now suppose phi: G -> G0 is a homomorphism of groups and N = Ker(phi) is its kernel.
(i) If H is a subgroup of G, show that HN = NH. (Warning: this is an equation of sets; proceed
accordingly; do not assume that G is abelian.)
(i

Please see the attached file for the fully formatted problems.
1 Prove that the solution space of AX = 0, where A is a m x n matrix, is a
vector space.
2 Are the vectors x3 - 1, x2 - x and x linearly independent in P3 ? Why ?
3 Determine whether or not the function T : Mmn --> Mmn dened by T(A) = A + B, where B is a mix

1) Define a linear transformation....
a) Find a basis for Ker T.
b) Find a basis for Im T.
c) Is T an onto map?
d) Is T a one-to-one map?
2) Define a linear transformation...
a) Find the matrix for T with respect to the standard basis.
b) Find the matrix for T with respect to { ( ) , ( ) , ( ) } as the basis for R and t

B1) This question concerns the following two subsets of :
(a) Show that , and find a vector in that does not belong to T. [3]
(b) Show that T is a subspace of . [4]
(c) Show that S is a basis for T, and write down the dimension of T. [7]
(d) Find an orthogonal basis for T that contains the vector .

10. Let A = a11 a12 a13 Show that A has rank 2 if and only if one or more of
a21 a22 a23 the determinants
a11 a12 a11 a13 a12 a13
a21 a22 a21 a23 a22 a23 are non zero.
14. Use the result in Exercise 10 to show that the set of points (x, y, z) in R3 for which the matrix

Suppose that G is a finite Abelian group and G has no element of order 2. Show that the mapping g-->g^2 is an automorphism of G. Show, by example, that if G is infinite the mapping need not be an automorphism (hint: consider Z).