###### Exercise 6

Determine whether the following statements are true or false and provide a justification for your response.

If the columns of a matrix \(A\) form a basis for \(\real^m\text{,}\) then \(A\) is invertible.

There must be 125 vectors in a basis for \(\real^{125}\text{.}\)

If \(\bcal=\{\vvec_1,\vvec_2,\ldots,\vvec_n\}\) is a basis of \(\real^m\text{,}\) then every vector in \(\real^m\) can be expressed as a linear combination of basis vectors.

The coordinates \(\coords{\xvec}{\bcal}\) are the weights that form \(\xvec\) as a linear combination of basis vectors.

If the basis vectors form the columns of the matrix \(C_{\bcal}\text{,}\) then \(\coords{\xvec}{\bcal} = C_{\bcal}\xvec\text{.}\)