Question

3. (3pts) Consider the \(3 \times 3\) matrices \(B=\left[\begin{array}{ccc}1 & 1 & 2 \\ -1 & 0 & 4 \\ 0 & 0 & 1\end{array}\right]\) and \(A=\left[\begin{array}{lll}\mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3}\end{array}\right]\), where \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\), and \(\mathrm{a}_{9}\) are the columns of \(A\). Let \(A B=\left[\begin{array}{lll}v_{1} & v_{2} & v_{3}\end{array}\right]\), where \(v_{1}, v_{2}\), and \(v_{3}\) are the columns of the product. Express a as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\), and \(\mathbf{v}_{3}\).

4. (3pts) Let \(T(x)=A x\) be the linear transformation given by

$$ A=\left[\begin{array}{cccc} 3 & -6 & 10 & 6 \\ 1 & -3 & 2 & 1 \\ 1 & 0 & 6 & 3 \end{array}\right] $$

Is \(T\) onto? Explain why or why not. Is \(T\) one-to-one? Explain why or why not.

5. (3pts) State whether the following statements are True or False.

a) If a matrix \(A\) has more columns than rows, then the linear transformation \(T(x)=A x\) must be onto.

b) If the columns of an \(n \times n\) matrix are linearly independent, then they span \(\mathbb{R}^{n}\).

c) If a matrix \(A\) can be row reduced to the identity matrix, then \(A\) must be invertible.

Answer 1