# TBIL Activities for Understanding Linear Algebra

## Section2.2Matrix multiplication and linear combinations

### Definition2.2.0.1.

The product of a matrix $$A$$ by a vector $$\xvec$$ will be the linear combination of the columns of $$A$$ using the components of $$\xvec$$ as weights.
If $$A$$ is an $$m\times n$$ matrix, then $$\xvec$$ must be an $$n$$-dimensional vector, and the product $$A\xvec$$ will be an $$m$$-dimensional vector. If
\begin{equation*} A=\left[\begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \ldots \amp \vvec_n \end{array}\right], \xvec = \left[\begin{array}{r} c_1 \\ c_2 \\ \vdots \\ c_n \end{array}\right], \end{equation*}
then
\begin{equation*} A\xvec = c_1\vvec_1 + c_2\vvec_2 + \ldots c_n\vvec_n\text{.} \end{equation*}

### Activity2.2.0.1.

Use the definition of matrix multiplication to find the product of a matrix and a vector.

#### (a)

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*}

#### (b)

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 $$A\xvec\text{?}$$

### Activity2.2.0.2.

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

### Activity2.2.0.3.

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 be able to explain why $$I$$ is called the identity matrix.

### Activity2.2.0.4.

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*}

#### (a)

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

#### (b)

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 $$A\xvec = \bvec\text{?}$$

### Activity2.2.0.5.The equation $$A\xvec = \bvec$$.

We can now relate a matrix equation to a system of equations, where the vector $$\xvec$$ is a vector whose coordinates are the variables in the system of equations.

#### (a)

Consider the linear system
\begin{equation*} \begin{alignedat}{4} 2x \amp {}+{} \amp y \amp {}-{} \amp 3z \amp {}={} \amp 4 \\ -x \amp {}+{} \amp 2y \amp {}+{} \amp z \amp {}={} \amp 3 \\ 3x \amp {}-{} \amp y \amp \amp \amp {}={} \amp -4 \\ \end{alignedat}\text{.} \end{equation*}
Identify the matrix $$A$$ and vector $$\bvec$$ to express this system in the form $$A\xvec = \bvec\text{.}$$

#### (b)

If $$A$$ and $$\bvec$$ are as below, write the linear system corresponding to the equation $$A\xvec=\bvec\text{.}$$
\begin{equation*} A = \left[\begin{array}{rrr} 3 \amp -1 \amp 0 \\ -2 \amp 0 \amp 6 \end{array} \right], \bvec = \left[\begin{array}{r} -6 \\ 2 \end{array} \right] \end{equation*}
and describe the solution space in set notation.



#### (c)

Describe the solution space (in set notation) of the equation
\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] \xvec = \left[\begin{array}{r} -1 \\ 1 \\ 5 \end{array} \right]\text{.} \end{equation*}



### Activity2.2.0.6.

Suppose $$A$$ is an $$m\times n$$ matrix. Give the statement that best describes the solution space of the equation $$A\xvec = \zerovec\text{.}$$
1. We can't determine anything about the solution space without the actual matrix $$A\text{.}$$
2. The only solution for $$A\xvec = \zerovec$$ is $$\xvec=\zerovec\text{.}$$
3. The system is always consistent, but we can't determine any solutions without knowing $$A\text{.}$$
4. The system is always consistent, and must contain $$\zerovec\text{,}$$ but may also contain additional solutions.

### Activity2.2.0.7.

Consider the matrices
\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 3 \amp 2 \\ -3 \amp 4 \amp -1 \\ \end{array}\right], B = \left[\begin{array}{rr} 3 \amp 0 \\ 1 \amp 2 \\ -2 \amp -1 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

Suppose we want to form the product $$AB\text{.}$$ Before computing, first make sure you can explain how you know this product exists and then give the dimensions of the resulting matrix.

#### (b)

Compute the product $$AB\text{.}$$

#### (c)

Sage can multiply matrices using the * operator. Define the matrices $$A$$ and $$B$$ in the Sage cell below and check your work by computing $$AB\text{.}$$



#### (d)

Are you able to form the matrix product $$BA\text{?}$$ If so, use the Sage cell above to find $$BA\text{.}$$ Is it generally true that $$AB = BA\text{?}$$

### Activity2.2.0.8.

Suppose we form the three matrices.
\begin{equation*} A = \left[\begin{array}{rr} 1 \amp 2 \\ 3 \amp -2 \\ \end{array}\right], B = \left[\begin{array}{rr} 0 \amp 4 \\ 2 \amp -1 \\ \end{array}\right], C = \left[\begin{array}{rr} -1 \amp 3 \\ 4 \amp 3 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

Compare what happens when you compute $$A(B+C)$$ and $$AB + AC\text{.}$$ State your finding as a general principle.



#### (b)

Compare the results of evaluating $$A(BC)$$ and $$(AB)C$$ and state your finding as a general principle.

### Activity2.2.0.9.

When we are dealing with real numbers, we know if $$a\neq 0$$ and $$ab = ac\text{,}$$ then $$b=c\text{.}$$ Define matrices
\begin{equation*} A = \left[\begin{array}{rr} 1 \amp 2 \\ -2 \amp -4 \\ \end{array}\right], B = \left[\begin{array}{rr} 3 \amp 0 \\ 1 \amp 3 \\ \end{array}\right], C = \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 2 \\ \end{array}\right]. \end{equation*}

#### (a)

Compute $$AB$$ and $$AC\text{.}$$



#### (b)

If $$AB = AC\text{,}$$ is it necessarily true that $$B = C\text{?}$$

### Activity2.2.0.10.

Again, with real numbers, we know that if $$ab = 0\text{,}$$ then either $$a = 0$$ or $$b=0\text{.}$$ Define
\begin{equation*} A = \left[\begin{array}{rr} 1 \amp 2 \\ -2 \amp -4 \\ \end{array}\right], B = \left[\begin{array}{rr} 2 \amp -4 \\ -1 \amp 2 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

Compute $$AB\text{.}$$



#### (b)

If $$AB = 0\text{,}$$ is it necessarily true that either $$A=0$$ or $$B=0\text{?}$$