Given the linear transformation: T(M)=[1002]M−M[1002] a) Find the matrix B of T w respect to the standard basis B of R2×2. b) Find bases of the image and kernel of B?

Question

Given the linear transformation:  T(M)=[1002]M−M[1002]  a) Find the matrix B of T w respect to the standard basis B of R2×2.  b) Find bases of the image and kernel of B?

Jasmine Herman · Accepted Answer

Step 1  Let  M=[abcd]  from the definition of T we have:  T(M)=[0−bc0]  So, representing M and T(M) as vectors in standard basis, T acts as:  [abcd]→[0−bc0]  a simple inspection shows that T is represented by the matrix:  [00000−10000100000]  now you can find image and kernel.

saennwegoyk · Accepted Answer

Step 1  In this case you will have to use a coordinatization of the space of matrices M2×2(R) as R4. The usual coordinatization goes via  [abcd]↦[a,b,c,d]  In this fashion, you will have to plug in four matrices:  [1000],[0100],[0010],[0001]  These matrices are the standard basis in M2×2(R), and note that they correspond to the canonical base  {(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)}  according to our coordinatization.  Each of the operations will give you as an answer 2×2 matrix, that you will have to transform in a vector using coordinatization again. The matrix of T with respect to B will then by a 4×4 matrix.  From there, I think you can calculate Kernel and Image... just have always in mind that your final answers have to be given in terms of matrices, so you will have to use again the coordinatization (this time, to transform vector of R4 in matrices 2×2).

Given the linear transformation: \(T(M) = \begin{bmatrix}1&0\\0&2\end{bmatrix}M -M\begin{bmatrix}1&0\\0&2\end{bmatrix}\) a) Find the matrix

Answered question

Answer & Explanation

New Questions in Linear algebra