###### Activity2.2.2Matrix-vector multiplication
1. Find the matrix product

\begin{equation*} \left[ \begin{array}{rrrr} 1 \amp 2 \amp 0 \amp -1 \\ 2 \amp 4 \amp -3 \amp -2 \\ -1 \amp -2 \amp 6 \amp 1 \\ \end{array} \right] \left[ \begin{array}{r} 3 \\ 1 \\ -1 \\ 1 \\ \end{array} \right]\text{.} \end{equation*}
2. Suppose that $$A$$ is the matrix

\begin{equation*} \left[ \begin{array}{rrr} 3 \amp -1 \amp 0 \\ 0 \amp -2 \amp 4 \\ 2 \amp 1 \amp 5 \\ 1 \amp 0 \amp 3 \\ \end{array} \right]\text{.} \end{equation*}

If $$A\xvec$$ is defined, what is the dimension of the vector $$\xvec$$ and what is the dimension of $$A\xvec\text{?}$$

3. A vector whose entries are all zero is denoted by $$\zerovec\text{.}$$ If $$A$$ is a matrix, what is the product $$A\zerovec\text{?}$$

4. Suppose that $$I = \left[\begin{array}{rrr} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right]$$ is the identity matrix and $$\xvec=\threevec{x_1}{x_2}{x_3}\text{.}$$ Find the product $$I\xvec$$ and explain why $$I$$ is called the identity matrix.

5. Suppose we write the matrix $$A$$ in terms of its columns as

\begin{equation*} A = \left[ \begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \ldots \amp \vvec_n \\ \end{array} \right]\text{.} \end{equation*}

If the vector $$\evec_1 = \left[\begin{array}{r} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right]\text{,}$$ what is the product $$A\evec_1\text{?}$$

6. Suppose that

\begin{equation*} A = \left[ \begin{array}{rrrr} 1 \amp 2 \\ -1 \amp 1 \\ \end{array} \right], \bvec = \left[ \begin{array}{r} 6 \\ 0 \end{array} \right]\text{.} \end{equation*}

Is there a vector $$\xvec$$ such that $$A\xvec = \bvec\text{?}$$

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