Dimension range rank theorem

Question

Let F: R3 → R3 be a linear transformation defined by:

F([ 1 0 0 ]) = [ 3 1 0 ] ,F([ 1 1 0 ]) = [ a 1 1 ] ,F([ 0 0 1 ]) = [ 0 0 1 ],

where a∈R.

(a) Find the values of F on all standard basis vectors in R³, showing working, and hence write the standard matrix of F.

(b) For what values of a is dim⁡R(F) = 2? Give justification for your answer.

Answer 1

F(1, 0, 0) = (3, 1, 0)

F(1, 1, 0) = (a, 1, 1)

F(0, 0, 1) = (0, 0, 1)

F(1, 1, 0) = F[1(1, 0, 0) + 1(0, 1, 0) + 0(0, 0, 1)]

F(1, 1, 0) = F(1, 0, 0) + F(0, 1, 0)

So, F(0, 1, 0) = F(1, 1, 0) - F(1, 0, 0) = (a, 1, 1) - (3, 1, 0) = (a-3, 0, 1)

The standard matrix, M , for F is the 3x3 matrix with first column F(1, 0, 0) = (3, 1, 0), second column

F(0, 1, 0) = (a-3, 0, 1) and third column F(0, 0, 1) = (0, 0, 1).

If a-3 = 0, then the second and third columns of M are the same, so there are only 2 linearly independent columns, which means that the column space is 2 dimensional. Since M is a square matrix, the row space and column space have the same dimension. So, the row space is also of dimension 2.