Section 3.3 - Linear Independence
Exercises
1.
\(\vec v_1, \vec v_2,\) and \(\vec v_3\) are not linearly independent.
2.
- 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.
4 pivot columns
4.
4 pivot columns
5.
No since there cannot be a pivot in every column.
6.
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.
\(\vec y_1, \vec y_2\), and \(\vec y_3\) are not linearly independent.
8.
The vectors \(x-1, x-2,\) and \(x^2 +1\) are linearly independent in \(P_3.\)
9.
The vectors \(1, \cos(x), \) and \(\sin(x)\) are linearly independent in \(C[0, 2\pi].\)