Section 3.3 - Linear Independence
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. Determine if the following vectors are linearly independent:
\[
\vec{v}_1 = \begin{bmatrix} 1 \\ 2 \\ 2 \\ -3 \end{bmatrix}, \qquad
\vec{v}_2 = \begin{bmatrix} 3 \\ 7 \\ 9 \\ -4 \end{bmatrix}, \qquad
\vec{v}_3 = \begin{bmatrix} -2 \\ -5 \\ -7 \\ 1 \end{bmatrix}
\]
\(\vec v_1, \vec v_2,\) and \(\vec v_3\) are not linearly independent.
2. Let \(\vec{u} = \begin{bmatrix} 3 \\ 2 \\ -4 \end{bmatrix}\), \(\vec{v}=\begin{bmatrix} -6 \\ 1 \\ 7 \end{bmatrix}\), \(\vec{w} = \begin{bmatrix} 0 \\ -5 \\ 2 \end{bmatrix}\), and \(\vec{z} = \begin{bmatrix} 3 \\ 7 \\ -5 \end{bmatrix}.\)
- Are the sets \(\{\vec{u},\vec{v}\}\), \(\{\vec{u},\vec{w}\}\), \(\{\vec{u},\vec{z}\}\), and \(\{\vec{w},\vec{z}\}\) each linearly independent? Why or why not?
- Is the set \(\{\vec{u}, \vec{v}, \vec{w}, \vec{z}\}\) linearly independent? Why or why not?
- Each of the sets are linearly independent since each pair of vectors are not multiples of one another.
- The set \(\{\vec u, \vec v, \vec w, \vec z\}\) is linearly dependent since the augmented matrix (shown below) has infinitely many solutions.
\[\left[\begin{array}{cccc|c}
3 & -6 & 0 & 3 & 0\\
2 & 1 & -5 & 7 & 0 \\
-4 & 7 & 2 & -5 & 0
\end{array}\right]\]
3. Suppose the columns of a \(6\times 4\) matrix \(A\) are linearly independent. How many pivot columns does \(A\) have?
4 pivot columns
4. Suppose the columns of a \(4\times 6\) matrix \(B\) span \(\mathbb{R}^4\). How many pivot columns does \(B\) have?
4 pivot columns
5. Can the columns of a \(4 \times 6\) matrix be linearly independent?
No since there cannot be a pivot in every column.
6. Construct \(3 \times 2\) matrices \(A\) and \(B\) such that \(A\vec{x}=\vec{0}\) has only the trivial solution and \(B\vec{x}=\vec{0}\) has a nontrivial solution.
Answers may vary, but essentially \(A\) should have a pivot in every column and \(B\) should have at least one column with no pivot. One possible answer is below.
\[A=\begin{bmatrix}
1 & 0 \\
0 & 1 \\
0 & 0
\end{bmatrix}, \quad
B = \begin{bmatrix}
1 & 0 \\
0 & 0 \\
0 & 0
\end{bmatrix}\]
\[A=\begin{bmatrix}
1 & 0 \\
0 & 1 \\
0 & 0
\end{bmatrix}, \quad
B = \begin{bmatrix}
1 & 0 \\
0 & 0 \\
0 & 0
\end{bmatrix}\]
7. Let \(\vec{x}_1\), \(\vec{x}_2\), and \(\vec{x}_3\) be linearly independent vectors in \(\mathbb{R}^3\) and let
\[
\vec{y}_1 = \vec{x}_2 - \vec{x}_1, \qquad \vec{y}_2=\vec{x_3} - \vec{x}_2, \qquad \vec{y}_3 = \vec{x}_1 - \vec{x}_3
\]
Are \(\vec{y}_1\), \(\vec{y}_2\), and \(\vec{y}_3\) linearly independent?
\(\vec y_1, \vec y_2\), and \(\vec y_3\) are not linearly independent.
8. Determine whether the vectors \(x-1\), \(x-2\), and \(x^2+1\) are linearly independent in \(P_3.\)
The vectors \(x-1, x-2,\) and \(x^2 +1\) are linearly independent in \(P_3.\)
9. Determine whether the vectors \(1\), \(\cos(x)\), and \(\sin(x)\) are linearly independent in \(C[0,2\pi].\)
The vectors \(1, \cos(x), \) and \(\sin(x)\) are linearly independent in \(C[0, 2\pi].\)