Exercise 6

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

1. If the columns of a matrix $A$ form a basis for $\real^m$, then $A$ is invertible.

2. There must be 125 vectors in a basis for $\real^{125}$.

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

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

5. If the basis vectors form the columns of the matrix $P_{\bcal}$, then $\coords{\xvec}{\bcal} = P_{\bcal}\xvec$.