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The Timpson Lyceum · Linear Algebra (2026)

1.7 Exercises

Problem sheet · 20 points · 4 extra credit

Linear Independence

  1. 1. (2 pts) Determine whether the columns of [1235]\begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix} are linearly independent.

  2. 2. (3 pts) Determine whether v1=[121]\mathbf{v}_1 = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, v2=[243]\mathbf{v}_2 = \begin{bmatrix} 2 \\ 4 \\ 3 \end{bmatrix}, v3=[120]\mathbf{v}_3 = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} are linearly independent. If they are dependent, give an explicit dependence relation.

  3. 3. (3 pts) Determine whether v1=[101]\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, v2=[210]\mathbf{v}_2 = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}, v3=[013]\mathbf{v}_3 = \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix} are linearly independent.

  4. 4. (3 pts) Find all values of hh for which the vectors [1−12]\begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}, [301]\begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix}, [h2−3]\begin{bmatrix} h \\ 2 \\ -3 \end{bmatrix} are linearly dependent.

  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 R3\mathbb{R}^3 is linearly dependent.

    (c) If {v1,v2,v3}\{\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.

  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) {[1234],[5678],[0000]}\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 R4\mathbb{R}^4.

    (c) {[100],[010]}\left\{\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\right\}

  7. 7. (3 pts) Prove that {v1,v2,v3}\{\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.

  8. 8. (4 pts ••) Suppose {v1,v2,v3}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} is linearly independent. Prove that {v1+v2,  v2+v3,  v1+v3}\{\mathbf{v}_1 + \mathbf{v}_2, \; \mathbf{v}_2 + \mathbf{v}_3, \; \mathbf{v}_1 + \mathbf{v}_3\} is also linearly independent.