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Linear Algebra for Team-Based Inquiry Learning

2022 Edition

Steven Clontz	Drew Lewis
University of South Alabama	University of South Alabama

August 2, 2022

Section 2.5: Linear Independence (VS5)

Activity 2.5.1 (~10 min)

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{1em} 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 2.5.2

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.

Figure 1. A linearly dependent set of three vectors

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 2.5.3 (~10 min)

Begin with 3 vectors in $\IR^3$

\begin{equation*} \vec v_1=\left[\begin{array}{c}1 \\ 0 \\ 0\end{array}\right], \vec v_2=\left[\begin{array}{c}0 \\ 1 \\ 0\end{array}\right], \text{ and } \vec v_3=\left[\begin{array}{c}0 \\ 0 \\ 1\end{array}\right]\text{.} \end{equation*}

Part 1.

Choose three non-zero scalars, $a,b\text{,}$ and $c\text{.}$ Let $\vec w = a\vec v_1 + b \vec v_2 + c \vec v_3\text{.}$ Is the set $\{\vec v_1,\vec v_2,\vec v_3,\vec w\}$ linearly dependent?

Activity 2.5.3 (~10 min)

Begin with 3 vectors in $\IR^3$

Part 2.

Find

\begin{equation*} \RREF \left[\begin{array}{cccc} \vdots & \vdots &\vdots &\vdots \\ \vec v_1 & \vec v_2 & \vec v_3 & \vec w \\ \vdots &\vdots &\vdots &\vdots \end{array}\right]. \end{equation*}

What does this tell you about solution set for the vector equation $x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 + x_4\vec w=\vec{0}\text{?}$

Activity 2.5.4 (~10 min)

Let $\vec{v}_1,\vec{v}_2,\vec{v}_3 $ be vectors in $\mathbb R^n\text{.}$ Suppose $3\vec{v}_1-5\vec{v}_2=\vec{v}_3\text{,}$ so the set $\{\vec{v}_1,\vec{v}_2,\vec{v}_3\}$ is linearly dependent. Which of the following is true of the vector equation $x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3=\vec{0}\text{?}$

It is consistent with one solution.
It is consistent with infinitely many solutions.
It is inconsistent.

Fact 2.5.5

For any vector space, the set $\{\vec v_1,\dots\vec v_n\}$ is linearly dependent if and only if the vector equation $x_1\vec v_1+\dots+x_n\vec v_n=\vec{0}$ is consistent with infinitely many solutions.

Activity 2.5.6 (~10 min)

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 2.5.7

A set of Euclidean 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 a column without a pivot position.

Observation 2.5.8

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 $\{\vec v_1,\dots\vec v_n\}$ 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 $\{\vec v_1,\dots\vec v_n\}$ is linearly independent if and only the vector equation
\begin{equation*} c_1\vec v_1+ c_2 \vec v_2 + \cdots + c_n\vec v_n = 0 \end{equation*}
has exactly one solution, with $c_1 = c_2 = \cdots = c_n = 0\text{.}$

Activity 2.5.9 (~5 min)

Consider whether the set of Euclidean vectors $\left\{ \left[\begin{array}{c}-4\\2\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}1\\2\\0\\0\\3\end{array}\right], \left[\begin{array}{c}1\\10\\10\\2\\6\end{array}\right], \left[\begin{array}{c}3\\4\\7\\2\\1\end{array}\right] \right\}$ is linearly dependent or linearly independent.

Part 1.

Reinterpret this question as an appropriate question about solutions to a vector equation.

Activity 2.5.9 (~5 min)

Part 2.

Use the solution to this question to answer the original question.

Activity 2.5.10 (~10 min)

Consider whether the set of polynomials $\left\{ x^3+1,x^2+2x,x^2+7x+4 \right\}$ is linearly dependent or linearly independent.

Part 1.

Reinterpret this question as an appropriate question about solutions to a polynomial equation.

Activity 2.5.10 (~10 min)

Consider whether the set of polynomials $\left\{ x^3+1,x^2+2x,x^2+7x+4 \right\}$ is linearly dependent or linearly independent.

Part 2.

Use the solution to this question to answer the original question.

Activity 2.5.11 (~5 min)

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 2.5.12 (~5 min)

What is the largest number of

\begin{equation*} \P_4=\setBuilder{ax^4+bx^3+cx^2+dx+e}{a,b,c,d,e\in\IR} \end{equation*}

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 2.5.13 (~5 min)

What is the largest number of

\begin{equation*} \P=\setBuilder{f(x)}{f(x)\text{ is any polynomial}} \end{equation*}

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 2.5.14

Is is possible for the set of vectors $\{\vec v_1, \vec v_2,\ldots, \vec v_n, \vec 0\}$ in a vector space $V $ to be linearly independent? Recall that $\vec 0$ represents the additive identity.