# TBIL Activities for Understanding Linear Algebra

## Section2.1Vectors and linear combinations

### Activity2.1.0.1.

Suppose that
\begin{equation*} \vvec = \left[\begin{array}{r} 3 \\ 1 \end{array} \right], \wvec = \left[\begin{array}{r} -1 \\ 2 \end{array} \right]. \end{equation*}

#### (a)

Find expressions for the vectors
\begin{equation*} \begin{array}{cccc} \vvec, \amp 2\vvec, \amp -\vvec, \amp -2\vvec, \\ \wvec, \amp 2\wvec, \amp -\wvec, \amp -2\wvec\text{.} \\ \end{array} \end{equation*}
and sketch them in the $$xy$$ plane.

#### (b)

What geometric effect does scalar multiplication by a positive number have on a vector? (A) preserves the length, changes the direction, (B) preserves the direction, changes the length, (C) changes the length and the direction.

#### (c)

What geometric effect does scalar multiplication by a negative number have on a vector? (A) preserves the length, changes the direction, (B) preserves the direction, changes the length, (C) changes the length and the direction.

### Activity2.1.0.2.

As in the previous activity, suppose that
\begin{equation*} \vvec = \left[\begin{array}{r} 3 \\ 1 \end{array} \right], \wvec = \left[\begin{array}{r} -1 \\ 2 \end{array} \right]. \end{equation*}

#### (a)

Sketch the vectors $$\vvec, \wvec, \vvec + \wvec$$ in the $$xy$$ plane.

#### (b)

Consider vectors that have the form $$\vvec + a\wvec$$ where $$a$$ is any scalar. Sketch a few of these vectors when, say, $$a = -2, -1, 0, 1,$$ and $$2\text{.}$$ Consider the points at the tips of each of these vectors. An appropriate geometric description for this set of points would be
1. a line through the tip of $$\vvec$$ parallel to $$\wvec\text{.}$$
2. a line through the tip of $$\wvec$$ parallel to $$\vvec\text{.}$$
3. a line through $$(0,0)$$ parallel to $$\vvec+\wvec\text{.}$$

### Activity2.1.0.3.

As in the previous activity, suppose that
\begin{equation*} \vvec = \left[\begin{array}{r} 3 \\ 1 \end{array} \right], \wvec = \left[\begin{array}{r} -1 \\ 2 \end{array} \right]. \end{equation*}

#### (a)

If $$a$$ and $$b$$ are two scalars, then the vector
\begin{equation*} a \vvec + b \wvec \end{equation*}
is called a linear combination of the vectors $$\vvec$$ and $$\wvec\text{.}$$ Find the vector that is the linear combination when $$a = -2$$ and $$b = 1\text{.}$$

#### (b)

Can the vector $$\left[\begin{array}{r} -31 \\ 37 \end{array}\right]$$ be represented as a linear combination of $$\vvec$$ and $$\wvec\text{?}$$

### Activity2.1.0.4.

In this activity, we will look at linear combinations of a pair of vectors,
\begin{equation*} \vvec = \left[\begin{array}{r} 2 \\ 1 \end{array}\right], \wvec = \left[\begin{array}{r} 1 \\ 2 \end{array}\right] \end{equation*}
with weights $$a$$ and $$b\text{.}$$
Go to the Applet in Activity 2.1.2 2  in Understanding Linear Algebra.

#### (a)

The weight $$b$$ is initially set to 0. Explain what happens as you vary $$a$$ with $$b=0\text{?}$$ How is this related to scalar multiplication?

#### (b)

What is the linear combination of $$\vvec$$ and $$\wvec$$ when $$a = 1$$ and $$b=-2\text{?}$$ You may find this result using the diagram, but you should also verify it by computing the linear combination.

#### (c)

The vectors that arise when the weight $$b$$ is set to 1 and $$a$$ is varied is
1. a line through $$\vvec$$ parallel to $$\wvec\text{.}$$
2. a line through $$\wvec$$ parallel to $$\vvec\text{.}$$
3. a line through $$(0,0)$$ parallel to $$\vvec+\wvec\text{.}$$

#### (d)

Can the vector $$\left[\begin{array}{r} 0 \\ 0 \end{array} \right]$$ be expressed as a linear combination of $$\vvec$$ and $$\wvec\text{?}$$ If so, what are weights $$a$$ and $$b\text{?}$$

#### (e)

Can the vector $$\left[\begin{array}{r} 3 \\ 0 \end{array} \right]$$ be expressed as a linear combination of $$\vvec$$ and $$\wvec\text{?}$$ If so, what are weights $$a$$ and $$b\text{?}$$

#### (f)

Verify the result from the previous part by algebraically finding the weights $$a$$ and $$b$$ that form the linear combination $$\left[\begin{array}{r} 3 \\ 0 \end{array} \right]\text{.}$$

#### (g)

Can the vector $$\left[\begin{array}{r} 1.3 \\ -1.7 \end{array} \right]$$ be expressed as a linear combination of $$\vvec$$ and $$\wvec\text{?}$$ What about the vector $$\left[\begin{array}{r} 15.2 \\ 7.1 \end{array} \right]\text{?}$$

#### (h)

Are there any two-dimensional vectors that cannot be expressed as linear combinations of $$\vvec$$ and $$\wvec\text{?}$$

### Activity2.1.0.5.

Given the vectors
\begin{equation*} \vvec_1 = \left[\begin{array}{r} 4 \\ 0 \\ 2 \\ 1 \end{array} \right], \vvec_2 = \left[\begin{array}{r} 1 \\ -3 \\ 3 \\ 1 \end{array} \right], \vvec_3 = \left[\begin{array}{r} -2 \\ 1 \\ 1 \\ 0 \end{array} \right], \bvec = \left[\begin{array}{r} 0 \\ 1 \\ 2 \\ -2 \end{array} \right]\text{,} \end{equation*}
we ask if $$\bvec$$ can be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{.}$$ Rephrase this question by writing a linear system for the weights $$c_1\text{,}$$ $$c_2\text{,}$$ and $$c_3$$ and use the Sage cell below to answer this question.



### Activity2.1.0.6.

Consider the following linear system.
\begin{equation*} \begin{alignedat}{4} 3x_1 \amp {}+{} \amp 2x_2 \amp {}-{} x_3 \amp {}={} \amp 4 \\ x_1 \amp \amp \amp {}+{} 2x_3 \amp {}={} \amp 0 \\ -x_1 \amp {}-{} \amp x_2 \amp {}+{} 3x_3 \amp {}={} \amp 1 \\ \end{alignedat} \end{equation*}
Identify vectors $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ $$\vvec_3\text{,}$$ and $$\bvec$$ and rephrase the question "Is this linear system consistent?" by asking "Can $$\bvec$$ be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{?}$$"

### Activity2.1.0.7.

Consider the vectors
\begin{equation*} \vvec_1 = \left[\begin{array}{r} 0 \\ -2 \\ 1 \\ \end{array} \right], \vvec_2 = \left[\begin{array}{r} 1 \\ 1 \\ -1 \\ \end{array} \right], \vvec_3 = \left[\begin{array}{r} 2 \\ 0 \\ -1 \\ \end{array} \right], \bvec = \left[\begin{array}{r} -1 \\ 3 \\ -1 \\ \end{array} \right]\text{.} \end{equation*}

#### (a)

Which statement is the most accurate for $$\bvec\text{?}$$
1. The vector $$\bvec$$ can be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ in exactly one way.
2. The vector $$\bvec$$ can be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ in more than one way.
3. The vector $$\bvec$$ cannot be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{.}$$



#### (b)

Considering just the vectors $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{,}$$ determine the most accurate statement.
1. Every three dimensional vector $$\bvec$$ can be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ in exactly one way.
2. Every three dimensional vector $$\bvec$$ can be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ in more than one way.
3. There are vectors $$\bvec$$ that cannot be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{.}$$
Be able to explain how the pivot positions of the matrix $$\left[\begin{array}{rrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \end{array} \right]$$ help answer this question.

### Activity2.1.0.8.

Consider the vectors
\begin{equation*} \vvec_1 = \left[\begin{array}{r} 0 \\ -2 \\ 1 \\ \end{array} \right], \vvec_2 = \left[\begin{array}{r} 1 \\ 1 \\ -1 \\ \end{array} \right], \vvec_3 = \left[\begin{array}{r} 1 \\ -1 \\ -2 \\ \end{array} \right], \bvec = \left[\begin{array}{r} 0 \\ 8 \\ -4 \\ \end{array} \right]\text{.} \end{equation*}

#### (a)

Which statement is the most accurate for $$\bvec\text{?}$$
1. The vector $$\bvec$$ can be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ in exactly one way.
2. The vector $$\bvec$$ can be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ in more than one way.
3. The vector $$\bvec$$ cannot be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{.}$$



#### (b)

Considering just the vectors $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ from the previous part, determine the most accurate statement.
1. Every three dimensional vector $$\bvec$$ can be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ in exactly one way.
2. Every three dimensional vector $$\bvec$$ can be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ in more than one way.
3. There are vectors $$\bvec$$ that cannot be expressed as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{.}$$
Be able to explain how the pivot positions of the matrix $$\left[\begin{array}{rrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \end{array} \right]$$ help answer this question.
davidaustinm.github.io/ula/sec-vectors-lin-combs.html#activity-11