Linear independence

Section 2.4 Linear independence

We start these activities by studying the span of two different sets of vectors.

Activity 2.4.0.1.

Consider the following vectors in \(\real^3\text{:}\)

\begin{equation*} \vvec_1=\threevec{0}{-1}{2}, \vvec_2=\threevec{3}{1}{-1}, \vvec_3=\threevec{2}{0}{1}\text{.} \end{equation*}

Describe the span of these vectors, \(\laspan{\vvec_1,\vvec_2,\vvec_3}\text{.}\)

Activity 2.4.0.2.

We will now consider a set of vectors that looks very much like the first set:

\begin{equation*} \wvec_1=\threevec{0}{-1}{2}, \wvec_2=\threevec{3}{1}{-1}, \wvec_3=\threevec{3}{0}{1}\text{.} \end{equation*}

(a)

Describe the span of these vectors, \(\laspan{\wvec_1,\wvec_2,\wvec_3}\text{.}\)

(b)

Show that the vector \(\wvec_3\) is a linear combination of \(\wvec_1\) and \(\wvec_2\) by finding weights such that

\begin{equation*} \wvec_3 = a\wvec_1 + b\wvec_2\text{.} \end{equation*}

(c)

Using your work from the previous part, show how any any linear combination of \(\wvec_1\text{,}\) \(\wvec_2\text{,}\) and \(\wvec_3\text{,}\)

\begin{equation*} c_1\wvec_1 + c_2\wvec_2 + c_3\wvec_3 \end{equation*}

can be written as a linear combination of \(\wvec_1\) and \(\wvec_2\text{.}\)

(d)

Explain why

\begin{equation*} \laspan{\wvec_1,\wvec_2,\wvec_3} = \laspan{\wvec_1,\wvec_2}\text{.} \end{equation*}

Definition 2.4.0.1.

A set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) is called linearly dependent if one of the vectors is a linear combination of the others. Otherwise, the set of vectors is called linearly independent.

We would like to develop a means of detecting when a set of vectors is linearly dependent. These questions will point the way.

Activity 2.4.0.3.

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.

(a)

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

(b)

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

Activity 2.4.0.4.

Suppose we have five vectors in \(\real^4\) that form the columns of a matrix having reduced row echelon form

\begin{equation*} \left[\begin{array}{rrrrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \amp \vvec_4 \amp \vvec_5 \end{array}\right] \sim \left[\begin{array}{rrrrr} 1 \amp 0 \amp -1 \amp 0 \amp 2 \\ 0 \amp 1 \amp 2 \amp 0 \amp 3 \\ 0 \amp 0 \amp 0 \amp 1 \amp -1 \\ 0 \amp 0 \amp 0 \amp 0 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}

(a)

Is it possible to write one of the vectors \(\vvec_1,\vvec_2,\ldots,\vvec_5\) as a linear combination of the others? If so, show explicitly how one vector appears as a linear combination of some of the other vectors.

(b)

Is this set of vectors linearly dependent or independent?

Activity 2.4.0.5.

Suppose we have another set of three vectors in \(\real^4\) that form the columns of a matrix having reduced row echelon form

\begin{equation*} \left[\begin{array}{rrr} \wvec_1 \amp \wvec_2 \amp \wvec_3 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ 0 \amp 0 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}

(a)

Is it possible to write one of these vectors \(\wvec_1\text{,}\) \(\wvec_2\text{,}\) \(\wvec_3\) as a linear combination of the others? If so, show explicitly how one vector appears as a linear combination of some of the other vectors.

(b)

Is this set of vectors linearly dependent or independent?

(c)

By looking at the pivot positions, how can you determine whether the columns of a matrix are linearly dependent or independent?

Activity 2.4.0.6.

If one vector in a set is the zero vector \(\zerovec\text{,}\) can the set of vectors be linearly independent?

Activity 2.4.0.7.

Suppose a set of vectors in \(\real^{10}\) has twelve vectors. Is it possible for this set to be linearly independent?

Activity 2.4.0.8.

Let \(\vvec_1,\vvec_2,\vvec_3 \) be vectors in \(\mathbb R^n\text{.}\) Suppose \(3\vvec_1-5\vvec_2=\vvec_3\text{,}\) so the set \(\{\vvec_1,\vvec_2,\vvec_3\}\) is linearly dependent. Which of the following is true of the vector equation \(x_1\vvec_1+x_2\vvec_2+x_3\vvec_3=\zerovec\text{?}\)

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

Activity 2.4.0.9.

We now want to explore the connection between linear independence and homogeneous equations.

(a)

Explain why the homogenous matrix equation \(A\xvec = \zerovec\) is consistent no matter the matrix \(A\text{.}\)

(b)

Consider the matrix

\begin{equation*} A = \left[\begin{array}{rrr} 3 \amp 2 \amp 0 \\ -1 \amp 0 \amp -2 \\ 2 \amp 1 \amp 1 \end{array}\right] \end{equation*}

whose columns we denote by \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\text{.}\) Are the vectors \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\) linearly dependent or independent?

(c)

Give a parametric description (in set notation) of the solution space of the homogeneous equation \(A\xvec = \zerovec\text{.}\)

(d)

We know that \(\zerovec\) is a solution to the homogeneous equation. Find another solution that is different from \(\zerovec\text{.}\) Use your solution to find weights \(c_1\text{,}\) \(c_2\text{,}\) and \(c_3\) such that

\begin{equation*} c_1\vvec_1 + c_2\vvec_2 + c_3\vvec_3 = \zerovec\text{.} \end{equation*}

(e)

Use the expression you found in the previous part to write one of the vectors as a linear combination of the others.

Activity 2.4.0.10.

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.

TBIL Activities for Understanding Linear Algebra

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