web analytics
Skip to main content

TBIL Activities for Understanding Linear Algebra

Jennifer Nordstrom

Section 3.2 Bases and coordinate systems

Activity 3.2.0.1.

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.

Definition 3.2.0.1.

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.

Activity 3.2.0.2.

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 \(\real^3\text{?}\)

Activity 3.2.0.3.

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?

Activity 3.2.0.4.

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*}

Activity 3.2.0.5.

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.

Activity 3.2.0.6.

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.

Activity 3.2.0.7.

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 \(\bcal\text{.}\)

Activity 3.2.0.8.

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
<Prev^TopNext>
Authored in PreTeXt
Powered by Runestone