Hi,
Consider the vector space M2×2 of all 2 × 2 matrices with real number entries. Let T be the linear transformation T : M2×2 > M2×2 given by:
T( ) = =
(a) Write down the standard basis S for M2×2 and write down the dimension of M2×2.
I found that S={(1,0),(0,1)} and the dimension of M2x2 is 4 (2x2=4)
(b) Write down the image under T of each element of S and hence write down the matrix of T with respect to S.
Not sure?
(c) Find a basis (consisting of 2 × 2 matrices) for the image of T and hence write down the rank and nullity of T.
Not sure?
(d) Find a basis (consisting of 2 × 2 matrices) for the nullspace of T.
Not sure?
Please help!
Did you think about this? If your basis contained only a single matrix, then the dimension would be 1, not 4. The "standard basis" for M2x2 is , , , and . That should have been one of the first things you learned.
(b) Write down the image under T of each element of S and hence write down the matrix of T with respect to S.
Not sure?
which is just the sum of 0 times each of the basis matrices. The first column is of the matrix representation of T is all 0s, .
so the second column is .
Now, try the next two columns.
(c) Find a basis (consisting of 2 × 2 matrices) for the image of T and hence write down the rank and nullity of T.Not sure?
Any vector in the image of T can be written as . Get the point?
(d) Find a basis (consisting of 2 × 2 matrices) for the nullspace of T.Any will be 0 (and so the matrix in the null space of T) if and only if b= 0 and d= 0. That leavesNot sure?
Please help!