Each of the matrices below represents a linear transformatio

Each of the matrices below represents a linear transformation from R^n to R^m. Determine the values of n and m for each matrix. Then determine their kernels and ranges and find a basis for each of these subspaces. [1 0 1] [1 0 1] [1 1 0 1] [1 1 1 1 0 1 1 1 1 1 0 -1]

Solution

a)

Transformation matrix is 1X3.

Then Rⁿ should be 1X1 and R^m should be 1X3.

So, n=1 and m=3.

Basis of Rⁿ = [1]

Basis of R^m = [1 0 0 ] , [0 1 0] , [0 0 1].