1.7 Exercises

20 points · 4 extra credit

Linear Independence

1. 1. (2 pts) Determine whether the columns of $\begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$ are linearly independent.

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2. 2. (3 pts) Determine whether $\mathbf{v}_1 = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$, $\mathbf{v}_2 = \begin{bmatrix} 2 \\ 4 \\ 3 \end{bmatrix}$, $\mathbf{v}_3 = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}$ are linearly independent. If they are dependent, give an explicit dependence relation.

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3. 3. (3 pts) Determine whether $\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$, $\mathbf{v}_2 = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}$, $\mathbf{v}_3 = \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix}$ are linearly independent.

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4. 4. (3 pts) Find all values of $h$ for which the vectors $\begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}$, $\begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix}$, $\begin{bmatrix} h \\ 2 \\ -3 \end{bmatrix}$ are linearly dependent.

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5. 5. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) A set of two vectors is linearly dependent if and only if one is a scalar multiple of the other.

   (b) Any set of four vectors in $\mathbb{R}^3$ is linearly dependent.

   (c) If $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent, then each of the three vectors is a linear combination of the other two.

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6. 6. (3 pts) Without any calculation, decide whether each set is linearly independent or dependent, and name the theorem or fact that justifies your answer.

   (a) $\left\{\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}, \begin{bmatrix} 5 \\ 6 \\ 7 \\ 8 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}\right\}$

   (b) A set of five vectors in $\mathbb{R}^4$.

   (c) $\left\{\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\right\}$

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7. 7. (3 pts) Prove that $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent if and only if at least one of the vectors is a linear combination of the other two.

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8. 8. (4 pts ••) Suppose $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly independent. Prove that $\{\mathbf{v}_1 + \mathbf{v}_2, \; \mathbf{v}_2 + \mathbf{v}_3, \; \mathbf{v}_1 + \mathbf{v}_3\}$ is also linearly independent.

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