Learning Outcomes
- Determine if a set of Euclidean vectors is linearly dependent or independent by solving an appropriate vector equation.
Section 2.4 Linear Independence (EV4)
Subsection 2.4.1 Class Activities
Activity 2.4.1.
Consider the two sets
\begin{equation*}
S=\left\{
\left[\begin{array}{c}2\\3\\1\end{array}\right],
\left[\begin{array}{c}1\\1\\4\end{array}\right]
\right\} \hspace{3em}
T=\left\{
\left[\begin{array}{c}2\\3\\1\end{array}\right],
\left[\begin{array}{c}1\\1\\4\end{array}\right],
\left[\begin{array}{c}-1\\0\\-11\end{array}\right]
\right\}\text{.}
\end{equation*}
Which of the following is true?
- \(\vspan S\) is bigger than \(\vspan T\text{.}\)
- \(\vspan S\) and \(\vspan T\) are the same size.
- \(\vspan S\) is smaller than \(\vspan T\text{.}\)
Definition 2.4.2.
We say that a set of vectors is linearly dependent if one vector in the set belongs to the span of the others. Otherwise, we say the set is linearly independent.
You can think of linearly dependent sets as containing a redundant vector, in the sense that you can drop a vector out without reducing the span of the set. In the above image, all three vectors lay in the same planar subspace, but only two vectors are needed to span the plane, so the set is linearly dependent.
Activity 2.4.3.
Consider the following three vectors in \(\IR^3\text{:}\)
\begin{equation*}
\vec v_1=\left[\begin{array}{c}-2 \\ 0 \\ 0\end{array}\right],
\vec v_2=\left[\begin{array}{c}1 \\ 3 \\ 0\end{array}\right],
\text{ and }
\vec v_3=\left[\begin{array}{c}-2 \\ 5 \\ 4\end{array}\right]\text{.}
\end{equation*}
(a)
Let \(\vec w = 3\vec v_1 - \vec v_2 - 5 \vec v_3 = \left[\begin{array}{c}\unknown \\ \unknown \\ \unknown\end{array}\right]\text{.}\) The set \(\{\vec v_1,\vec v_2,\vec v_3,\vec w\}\) is...
- linearly dependent: at least one vector is a linear combination of others
- linearly independent: no vector is a linear combination of others
(b)
Find
\begin{equation*}
\RREF \left[\begin{array}{cccc}
\vec v_1 & \vec v_2 & \vec v_3 & \vec w \\
\end{array}\right]=
\RREF \left[\begin{array}{cccc}
-2 & 1 &-2 & \unknown \\
0 & 3 & 5 & \unknown \\
0 &0 &4 & \unknown
\end{array}\right]= \unknown .
\end{equation*}
What does this tell you about solution set for the vector equation \(x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 + x_4\vec w=\vec{0}\text{?}\)
- It is inconsistent.
- It is consistent with one solution.
- It is consistent with infinitely many solutions.
(c)
Which of these might explain the connection?
- A pivot column establishes linear independence and creates a contradiction.
- A non-pivot column both describes a linear combination and reveals the number of solutions.
- A pivot row describes the bound variables and prevents a contradiction.
- A non-pivot row prevents contradictions and makes the vector equation solvable.
Fact 2.4.4.
For any vector space, the set \(\{\vec v_1,\dots\vec v_n\}\) is linearly dependent if and only if the vector equation \(x_1\vec v_1+ x_2 \vec v_2+\dots+x_n\vec v_n=\vec{0}\) is consistent with infinitely many solutions.
Likewise, the set of vectors \(\{\vec v_1,\dots\vec v_n\}\) is linearly independent if and only the vector equation
\begin{equation*}
x_1\vec v_1+ x_2 \vec v_2 + \cdots + x_n\vec v_n = 0
\end{equation*}
has exactly one solution: \(\left[\begin{array}{c}
x_1 \\ \vdots \\ x_n
\end{array}\right]=\left[\begin{array}{c}
0 \\ \vdots \\ 0
\end{array}\right]\text{.}\)
Activity 2.4.5.
Find
\begin{equation*}
\RREF\left[\begin{array}{ccccc|c}
2&2&3&-1&4&0\\
3&0&13&10&3&0\\
0&0&7&7&0&0\\
-1&3&16&14&1&0
\end{array}\right]
\end{equation*}
and mark the part of the matrix that demonstrates that
\begin{equation*}
S=\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\\1\end{array}\right]
\right\}
\end{equation*}
is linearly dependent (the part that shows its linear system has infinitely many solutions).
Observation 2.4.6.
Compare the following results:
- A set of \(\IR^m\) vectors \(\{\vec v_1,\dots\vec v_n\}\) is linearly independent if and only if \(\RREF\left[\begin{array}{ccc}\vec v_1&\dots&\vec v_n\end{array}\right]\) has all pivot columns.
- A set of \(\IR^m\) vectors \(\{\vec v_1,\dots\vec v_n\}\) is linearly dependent if and only if \(\RREF\left[\begin{array}{ccc}\vec v_1&\dots&\vec v_n\end{array}\right]\) has at least one non-pivot column.
- A set of \(\IR^m\) vectors \(\{\vec v_1,\dots\vec v_n\}\) spans \(\IR^m\) if and only if \(\RREF\left[\begin{array}{ccc}\vec v_1&\dots&\vec v_n\end{array}\right]\) has all pivot rows.
- A set of \(\IR^m\) vectors \(\{\vec v_1,\dots\vec v_n\}\) fails to span \(\IR^m\) if and only if \(\RREF\left[\begin{array}{ccc}\vec v_1&\dots&\vec v_n\end{array}\right]\) has at least one non-pivot row.
Activity 2.4.7.
(a)
Write a statement involving the solutions of a vector equation that’s equivalent to each claim:
(i)
“The set of vectors \(\left\{ \left[\begin{array}{c}
1 \\
-1 \\
0 \\
-1
\end{array}\right] , \left[\begin{array}{c}
5 \\
5 \\
3 \\
1
\end{array}\right] , \left[\begin{array}{c}
9 \\
11 \\
6 \\
3
\end{array}\right] \right\}\) is linearly independent.”
(ii)
“The set of vectors \(\left\{ \left[\begin{array}{c}
1 \\
-1 \\
0 \\
-1
\end{array}\right] , \left[\begin{array}{c}
5 \\
5 \\
3 \\
1
\end{array}\right] , \left[\begin{array}{c}
9 \\
11 \\
6 \\
3
\end{array}\right] \right\}\) is linearly dependent.”
(b)
Explain how to determine which of these statements is true.
Activity 2.4.8.
What is the largest number of \(\IR^4\) vectors that can form a linearly independent set?
- \(\displaystyle 3\)
- \(\displaystyle 4\)
- \(\displaystyle 5\)
- You can have infinitely many vectors and still be linearly independent.
Activity 2.4.9.
Is is possible for the set of Euclidean vectors \(\{\vec v_1, \vec v_2,\ldots, \vec v_n, \vec 0\}\) to be linearly independent?
- Yes
- No
Subsection 2.4.2 Videos
Exercises 2.4.3 Exercises
Exercises available at
https://teambasedinquirylearning.github.io/linear-algebra/2024e/exercises/#/bank/EV4/.Subsection 2.4.4 Mathematical Writing Explorations
Exploration 2.4.10.
Prove the result of Observation 2.4.6, by showing that, given a set \(S = \{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\}\) of vectors, \(S\) is linearly independent iff the equation \(x_1\vec{v}_1 + x_2\vec{v}_2 + \ldots\ + x_n\vec{v}_n = \vec{0}\) is only true when \(x_1 = x_2 = \cdots = x_n = 0\text{.}\)Subsection 2.4.5 Sample Problem and Solution
Sample problem Example B.1.8.
