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Section 3.2 Bases and coordinate systems
Activity 3.2.1 .
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 3.2.1 .
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 3.2.2 .
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 3.2.3 .
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 3.2.4 .
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 3.2.5 .
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 3.2.6 .
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 3.2.7 .
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 3.2.8 .
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.
Activity 3.2.9 .
