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The Timpson Lyceum · Linear Algebra (2026)

1.3 Exercises

Problem sheet · 20 points · 4 extra credit

Vector Equations

  1. 1. (2 pts) Let u=[2−13]\mathbf{u} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix} and v=[10−2]\mathbf{v} = \begin{bmatrix} 1 \\ 0 \\ -2 \end{bmatrix}. Compute 3u−2v3\mathbf{u} - 2\mathbf{v}.

  2. 2. (3 pts) Determine whether b=[53]\mathbf{b} = \begin{bmatrix} 5 \\ 3 \end{bmatrix} is a linear combination of a1=[12]\mathbf{a}_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix} and a2=[3−1]\mathbf{a}_2 = \begin{bmatrix} 3 \\ -1 \end{bmatrix}. If so, give the weights.

  3. 3. (3 pts) Determine whether b=[32−1]\mathbf{b} = \begin{bmatrix} 3 \\ 2 \\ -1 \end{bmatrix} lies in Span⁡{a1,a2,a3}\operatorname{Span}\{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\}, where

    a1=[10−1],a2=[210],a3=[012].\mathbf{a}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix},\quad \mathbf{a}_2 = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix},\quad \mathbf{a}_3 = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}.

  4. 4. (3 pts) Describe geometrically, in R3\mathbb{R}^3:

    (a) Span⁡{[12−1]}\operatorname{Span}\left\{\begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}\right\}

    (b) Span⁡{[100],[010]}\operatorname{Span}\left\{\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\right\}

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

    (a) The vector equation x1a1+x2a2+x3a3=bx_1\mathbf{a}_1 + x_2\mathbf{a}_2 + x_3\mathbf{a}_3 = \mathbf{b} has the same solution set as the linear system whose augmented matrix is [ a1  a2  a3∣b ][\,\mathbf{a}_1 \; \mathbf{a}_2 \; \mathbf{a}_3 \mid \mathbf{b}\,].

    (b) The span of a single nonzero vector in R3\mathbb{R}^3 is a plane.

    (c) Asking whether b∈Span⁡{a1,…,ap}\mathbf{b} \in \operatorname{Span}\{\mathbf{a}_1, \dots, \mathbf{a}_p\} is the same as asking whether the system [ a1  ⋯  ap∣b ][\,\mathbf{a}_1 \; \cdots \; \mathbf{a}_p \mid \mathbf{b}\,] is consistent.

  6. 6. (3 pts) Let a1=[1−2]\mathbf{a}_1 = \begin{bmatrix} 1 \\ -2 \end{bmatrix} and a2=[−36]\mathbf{a}_2 = \begin{bmatrix} -3 \\ 6 \end{bmatrix}. Describe exactly which vectors b=[b1b2]\mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} belong to Span⁡{a1,a2}\operatorname{Span}\{\mathbf{a}_1, \mathbf{a}_2\}, and give the defining equation of that set.

  7. 7. (3 pts) Explain why 0\mathbf{0} belongs to Span⁡{v1,…,vp}\operatorname{Span}\{\mathbf{v}_1, \dots, \mathbf{v}_p\} for any vectors v1,…,vp\mathbf{v}_1, \dots, \mathbf{v}_p. Then explain, for two vectors v1,v2\mathbf{v}_1, \mathbf{v}_2 in R3\mathbb{R}^3, the geometric possibilities for Span⁡{v1,v2}\operatorname{Span}\{\mathbf{v}_1, \mathbf{v}_2\}.

  8. 8. (4 pts ••) Let u\mathbf{u} and v\mathbf{v} be vectors in Rn\mathbb{R}^n. Prove that Span⁡{u,v}=Span⁡{u,u+v}\operatorname{Span}\{\mathbf{u}, \mathbf{v}\} = \operatorname{Span}\{\mathbf{u}, \mathbf{u} + \mathbf{v}\}.