# TBIL Activities for Understanding Linear Algebra

## Section3.2Bases and coordinate systems

### Activity3.2.0.1.

Consider the vectors
\begin{equation*} \vvec_1 = \twovec{2}{1}, \vvec_2 = \twovec{1}{2} \end{equation*}
in $$\real^2\text{.}$$

#### (a)

Find the linear combination $$\vvec_1 - 2\vvec_2\text{.}$$

#### (b)

Express the vector $$\twovec{-3}{0}$$ as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

#### (c)

Find the linear combination $$10\vvec_1 - 13\vvec_2\text{.}$$

#### (d)

Express the vector $$\twovec{16}{-4}$$ as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

#### (e)

Explain why every vector in $$\real^2$$ can be written as a linear combination of $$\vvec_1$$ and $$\vvec_2$$ in exactly one way.

### Definition3.2.0.1.

A set of vectors $$\vvec_1,\vvec_2,\ldots,\vvec_n$$ in $$\real^m$$ is called a basis for $$\real^m$$ if the set of vectors spans $$\real^m$$ and is linearly independent.

### Activity3.2.0.2.

Use the definition of a basis to determine whther the following sets of vectors form a basis for the appropriate $$\real^n$$

#### (a)

In the first activity, we considered a set of vectors in $$\real^2\text{:}$$
\begin{equation*} \vvec_1 = \twovec{2}{1}, \vvec_2 = \twovec{1}{2}\text{.} \end{equation*}
Explain why these vectors form a basis for $$\real^2\text{.}$$

#### (b)

Consider the set of vectors in $$\real^3$$
\begin{equation*} \vvec_1 = \threevec{1}{1}{1}, \vvec_2 = \threevec{0}{1}{-1}, \vvec_3 = \threevec{1}{0}{-1}\text{.} \end{equation*}
and determine whether they form a basis for $$\real^3\text{.}$$

#### (c)

Do the vectors
\begin{equation*} \vvec_1 = \threevec{-2}{1}{3}, \vvec_2 = \threevec{3}{0}{-1}, \vvec_3 = \threevec{1}{1}{0}, \vvec_4 = \threevec{0}{3}{-2} \end{equation*}
form a basis for $$\real^3\text{?}$$

### Activity3.2.0.3.

We can determine whether a set of vectors is a basis using RREF.

#### (a)

Explain why the vectors $$\evec_1,\evec_2,\evec_3$$ form a basis for $$\real^3\text{.}$$

#### (b)

If a set of vectors $$\vvec_1,\vvec_2,\ldots,\vvec_n$$ forms a basis for $$\real^m\text{,}$$ what can you guarantee about the pivot positions of the matrix
\begin{equation*} \left[\begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \ldots \amp \vvec_n \end{array}\right]\text{?} \end{equation*}

#### (c)

If the set of vectors $$\vvec_1,\vvec_2,\ldots,\vvec_n$$ is a basis for $$\real^{10}\text{,}$$ how many vectors must be in the set?

### Activity3.2.0.4.

If $$\{\vvec_1,\vvec_2,\vvec_3,\vvec_4\}$$ is a basis for $$\IR^4\text{,}$$ that means the reduced row echelon form of the matrix $$[\vvec_1\,\vvec_2\,\vvec_3\,\vvec_4]$$ doesn't have a column without a pivot position, and doesn't have a row of zeros. Determine the reduced row echelon form below.
\begin{equation*} \RREF[\vec v_1\,\vec v_2\,\vec v_3\,\vec v_4] = \left[\begin{array}{cccc} \unknown & \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown & \unknown \\ \end{array}\right] \end{equation*}

### Activity3.2.0.5.

Label each of the sets $$A,B,C,D,E$$ as
• SPANS $$\IR^4$$ or DOES NOT SPAN $$\IR^4$$
• LINEARLY INDEPENDENT or LINEARLY DEPENDENT
• BASIS FOR $$\IR^4$$ or NOT A BASIS FOR $$\IR^4$$
by finding $$\RREF$$ for their corresponding matrices.
\begin{align*} A&=\left\{ \left[\begin{array}{c}1\\0\\0\\0\end{array}\right], \left[\begin{array}{c}0\\1\\0\\0\end{array}\right], \left[\begin{array}{c}0\\0\\1\\0\end{array}\right], \left[\begin{array}{c}0\\0\\0\\1\end{array}\right] \right\} & B&=\left\{ \left[\begin{array}{c}2\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}2\\0\\0\\3\end{array}\right], \left[\begin{array}{c}4\\3\\0\\2\end{array}\right], \left[\begin{array}{c}-3\\0\\1\\3\end{array}\right] \right\}\\ C&=\left\{ \left[\begin{array}{c}2\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}2\\0\\0\\3\end{array}\right], \left[\begin{array}{c}3\\13\\7\\16\end{array}\right], \left[\begin{array}{c}-1\\10\\7\\14\end{array}\right], \left[\begin{array}{c}4\\3\\0\\2\end{array}\right] \right\} & D&=\left\{ \left[\begin{array}{c}2\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}4\\3\\0\\2\end{array}\right], \left[\begin{array}{c}-3\\0\\1\\3\end{array}\right], \left[\begin{array}{c}3\\6\\1\\5\end{array}\right] \right\}\\ E&=\left\{ \left[\begin{array}{c}5\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}-2\\1\\0\\3\end{array}\right], \left[\begin{array}{c}4\\5\\1\\3\end{array}\right] \right\} \end{align*}
Now we want to look at how to convert between two different bases. This is called change of basis.

### Activity3.2.0.6.

Let's begin with the basis $$\bcal = \{\vvec_1,\vvec_2\}$$ of $$\real^2$$ where
\begin{equation*} \vvec_1 = \twovec{3}{-2}, \vvec_2 = \twovec{2}{1}\text{.} \end{equation*}

#### (a)

If the coordinates of $$\xvec$$ in the basis $$\bcal$$ are $$\coords{\xvec}{\bcal} = \twovec{-2}{4}\text{,}$$ what is the vector $$\xvec\text{?}$$

#### (b)

If $$\xvec = \twovec{3}{5}\text{,}$$ find the coordinates of $$\xvec$$ in the basis $$\bcal\text{;}$$ that is, find $$\coords{\xvec}{\bcal}\text{.}$$

#### (c)

Find a matrix $$A$$ such that, for any vector $$\xvec\text{,}$$ we have $$\xvec = A\coords{\xvec}{\bcal}\text{.}$$ Explain why this matrix is invertible.

#### (d)

Using what you found in the previous part, find a matrix $$B$$ such that, for any vector $$\xvec\text{,}$$ we have $$\coords{\xvec}{\bcal} = B\xvec\text{.}$$ What is the relationship between the two matrices you have found in this and the previous part? Explain why this relationship holds.

### Activity3.2.0.7.

Suppose now we also consider the basis
\begin{equation*} \ccal = \left\{\twovec{1}{2}, \twovec{-2}{1}\right\}\text{.} \end{equation*}
Find a matrix $$M$$ that converts coordinates in the basis $$\ccal$$ into coordinates in the basis $$\bcal$$ from the previous activity; that is,
\begin{equation*} \coords{\xvec}{\bcal} = M \coords{\xvec}{\ccal}\text{.} \end{equation*}
You may wish to think about converting coordinates from the basis $$\ccal$$ into the standard coordinate system and then into the basis $$\bcal\text{.}$$

### Activity3.2.0.8.

Suppose we consider the standard basis
\begin{equation*} \ecal = \{\evec_1,\evec_2\}\text{.} \end{equation*}
What is the relationship between $$\xvec$$ and $$\coords{\xvec}{\ecal}\text{?}$$
In the next activity we explore an application of change of basis in the field of computer vision.
davidaustinm.github.io/ula/sec-bases.html#activity-32