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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(\(\displaystyle \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)) = \(\displaystyle \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) \(\displaystyle \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}\) = \(\displaystyle \begin{bmatrix} b & b \\ d & d \end{bmatrix}\)

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:

**I found that S={(1,0),(0,1)} and the dimension of M2x2 is 4 (2x2=4)**

(a) Write down the standard basis S for M2×2 and write down the dimension of M2×2.

(a) Write down the standard basis S for M2×2 and write down the dimension of M2×2.

**(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?

**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.

(c) Find a basis (consisting of 2 × 2 matrices) for the image of T and hence write down the rank and nullity of T.

(d) Find a basis (consisting of 2 × 2 matrices) for the nullspace of T.

(d) Find a basis (consisting of 2 × 2 matrices) for the nullspace of T.

Not sure?

Please help!