Let T be left multiplication by the matrix

  ⎡1  2  0   -1  5⎤
  ⎢               ⎥
  ⎢2  0  2   0   1⎥
  ⎢               ⎥.
  ⎢1  1  -1  3   2⎥
  ⎢               ⎥
  ⎣0  3  -3  2   6⎦

Compute ker T and im T explicitly by exhibiting bases for these spaces.

Answer

2 dimensional basis of ker T in \(F^5\), such as:

  ⎡-7⎤   ⎡0 ⎤
  ⎢  ⎥   ⎢  ⎥
  ⎢5 ⎥   ⎢-5⎥
  ⎢  ⎥   ⎢  ⎥
  ⎢7 ⎥ , ⎢-1⎥
  ⎢  ⎥   ⎢  ⎥
  ⎢3 ⎥   ⎢0 ⎥
  ⎢  ⎥   ⎢  ⎥
  ⎣0 ⎦   ⎣2 ⎦

3 dimensional basis of im T in \(F^4\), such as:

  ⎡1⎤   ⎡0⎤   ⎡0⎤
  ⎢ ⎥   ⎢ ⎥   ⎢ ⎥
  ⎢0⎥   ⎢1⎥   ⎢0⎥
  ⎢ ⎥ , ⎢ ⎥ , ⎢ ⎥
  ⎢0⎥   ⎢0⎥   ⎢1⎥
  ⎢ ⎥   ⎢ ⎥   ⎢ ⎥
  ⎣0⎦   ⎣0⎦   ⎣0⎦

Note T:V \(\rightarrow\) W, where dim(V) = 5 and dim(W) = 4 in this case; and

    dim(ker T) + dim(im T) = dim(V) 

in general per the Dimensional Formula (Ch 4.1).

Command line:

$ python --version
Python 3.7.0

$ python -m pip install --user sympy

Octave:

>> pkg load symbolic

>> A=[1 2 0 -1 5; 2 0 2 0 1; 1 1 -1 3 2; 0 3 -3 2 6]
A =

   1   2   0  -1   5
   2   0   2   0   1
   1   1  -1   3   2
   0   3  -3   2   6

>> A=sym(A)
A = (sym 4×5 matrix)

  ⎡1  2  0   -1  5⎤
  ⎢               ⎥
  ⎢2  0  2   0   1⎥
  ⎢               ⎥
  ⎢1  1  -1  3   2⎥
  ⎢               ⎥
  ⎣0  3  -3  2   6⎦

>> rref(A)
ans = (sym 4×5 matrix)

  ⎡1  0  0  7/3    0 ⎤
  ⎢                  ⎥
  ⎢0  1  0  -5/3  5/2⎥
  ⎢                  ⎥
  ⎢0  0  1  -7/3  1/2⎥
  ⎢                  ⎥
  ⎣0  0  0   0     0 ⎦