## Section2.3Spanning Sets (VS3)

### Subsection2.3.1Class Activities

#### Observation2.3.1.

Any single non-zero vector/number $$x$$ in $$\IR^1$$ spans $$\IR^1\text{,}$$ since $$\IR^1=\setBuilder{cx}{c\in\IR}\text{.}$$ Figure 8. An $$\IR^1$$ vector

#### Activity2.3.2.

How many vectors are required to span $$\IR^2\text{?}$$ Sketch a drawing in the $$xy$$ plane to support your answer. Figure 9. The $$xy$$ plane $$\IR^2$$
1. $$\displaystyle 1$$

2. $$\displaystyle 2$$

3. $$\displaystyle 3$$

4. $$\displaystyle 4$$

5. Infinitely Many

#### Activity2.3.3.

How many vectors are required to span $$\IR^3\text{?}$$ Figure 10. $$\IR^3$$ space
1. $$\displaystyle 1$$

2. $$\displaystyle 2$$

3. $$\displaystyle 3$$

4. $$\displaystyle 4$$

5. Infinitely Many

#### Activity2.3.5.

Choose any vector $$\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right]$$ in $$\IR^3$$ that is not in $$\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right], \left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}$$ by using technology to verify that $$\RREF \left[\begin{array}{cc|c}1&-2&\unknown\\-1&0&\unknown\\0&1&\unknown\end{array}\right] = \left[\begin{array}{cc|c}1&0&0\\0&1&0\\0&0&1\end{array}\right] \text{.}$$ (Why does this work?)

#### Activity2.3.7.

Consider the set of vectors $$S=\left\{ \left[\begin{array}{c}2\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}1\\-4\\3\\0\end{array}\right], \left[\begin{array}{c}1\\7\\-3\\-1\end{array}\right], \left[\begin{array}{c}0\\3\\5\\7\end{array}\right], \left[\begin{array}{c}3\\13\\7\\16\end{array}\right] \right\}$$ and the question “Does $$\IR^4=\vspan S\text{?}$$

##### (a)

Rewrite this question in terms of the solutions to a vector equation.

#### Activity2.3.8.

Consider the set of third-degree polynomials

\begin{align*} S=\{ &2x^3+3x^2-1, 2x^3+3, 3x^3+13x^2+7x+16,\\ &-x^3+10x^2+7x+14, 4x^3+3x^2+2 \} . \end{align*}

and the question “Does $$\P_3=\vspan S\text{?}$$

##### (a)

Rewrite this question to be about the solutions to a polynomial equation.

#### Activity2.3.9.

Consider the set of matrices

\begin{equation*} S = \left\{ \left[\begin{array}{cc} 1 & 3 \\ 0 & 1 \end{array}\right], \left[\begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array}\right], \left[\begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array}\right] \right\} \end{equation*}

and the question “Does $$M_{2,2} = \vspan S\text{?}$$

##### (a)

Rewrite this as a question about the solutions to a matrix equation.

#### Activity2.3.10.

Let $$\vec{v}_1, \vec{v}_2, \vec{v}_3 \in \IR^7$$ be three vectors, and suppose $$\vec{w}$$ is another vector with $$\vec{w} \in \vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\}\text{.}$$ What can you conclude about $$\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{?}$$

1. $$\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\}$$ is larger than $$\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}$$

2. $$\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} = \vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}$$

3. $$\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\}$$ is smaller than $$\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}$$

### Subsection2.3.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/VS3.slides.html.

### Subsection2.3.5Mathematical Writing Explorations

#### Exploration2.3.11.

Construct each of the following, or show that it is impossible:
• A set of 2 vectors that spans $$\mathbb{R}^3$$

• A set of 3 vectors that spans $$\mathbb{R}^3$$

• A set of 3 vectors that does not span $$\mathbb{R}^3$$

• A set of 4 vectors that spans $$\mathbb{R}^3$$

For any of the sets you constructed that did span the required space, are any of the vectors a linear combination of the others in your set?

#### Exploration2.3.12.

Based on these results, generalize this a conjecture about how a set of $$n-1, n$$ and $$n+1$$ vectors would or would not span $$\mathbb{R}^n\text{.}$$

### Subsection2.3.6Sample Problem and Solution

Sample problem Example B.1.7.