The Timpson Lyceum · Linear Algebra (2026)

4.3 Exercises

Problem sheet · 23 points · 4 extra credit

Linearly Independent Sets; Bases

1. (2 pts) Determine whether $\{(1,2),\,(2,4),\,(3,1)\}$ is linearly independent in $\mathbb{R}^2$ . Justify your answer.
2. (2 pts) Determine whether $\{(1,-1,2),\,(2,1,0),\,(0,3,-4)\}$ is linearly independent in $\mathbb{R}^3$ . If it is dependent, exhibit an explicit dependence relation.
3. (3 pts) Find a basis for the subspace $H = \text{Span}\,\{(1,0,-1),\,(2,1,1),\,(0,1,3),\,(1,1,2)\}$ of $\mathbb{R}^3$ .
4. (2 pts) Determine whether $\{1 + t,\; t + t^2,\; 1 + t^2\}$ is a basis for $P_2$ .
5. (3 pts) Show that $\{(1,1,0),\,(0,1,1)\}$ is linearly independent, then extend it to a basis for $\mathbb{R}^3$ by adjoining one more vector. Justify that your enlarged set is a basis.
6. (3 pts) State whether each of the following is true or false, and justify each answer.

(a) A set consisting of a single nonzero vector is always linearly independent.

(b) If a set of vectors contains the zero vector, it is linearly dependent.

(c) A basis for a vector space is a spanning set that is as large as possible.

(d) If $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ spans $V$ , then it is automatically a basis for $V$ .
7. (3 pts) Suppose $\mathbf{v}_3 = c_1\mathbf{v}_1 + c_2\mathbf{v}_2$ for scalars $c_1, c_2$ . Prove that $\text{Span}\,\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} = \text{Span}\,\{\mathbf{v}_1, \mathbf{v}_2\}$ .
8. (2 pts) Determine whether $\{(1,2,0),\,(0,1,3),\,(2,0,1)\}$ is a basis for $\mathbb{R}^3$ .
9. (3 pts) Let $A = \begin{bmatrix} 1 & 3 & 1 & 1 \\ 2 & 6 & 4 & 0 \\ 1 & 3 & 5 & -3 \end{bmatrix}.$ Find a basis for $\text{Nul}\,A$ by solving $A\mathbf{x} = \mathbf{0}$ , writing the general solution in parametric vector form and reading off the spanning vectors. Confirm that your basis vectors are linearly independent.
10. (4 pts ••) Prove that any set of $p$ vectors in $\mathbb{R}^n$ is linearly dependent whenever $p > n$ .