4.3 Exercises

23 points · 4 extra credit

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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.
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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.
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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 $\mathbb{R}^3$ .
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4. (2 pts) Determine whether $\{1 + t,\; t + t^2,\; 1 + t^2\}$ is a basis for $P_2$ .
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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.
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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 $V$ .
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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 $\text{Span}\,\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} = \text{Span}\,\{\mathbf{v}_1, \mathbf{v}_2\}$ .
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8. (2 pts) Determine whether $\{(1,2,0),\,(0,1,3),\,(2,0,1)\}$ is a basis for $\mathbb{R}^3$ .
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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.
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10. (4 pts ••) Prove that any set of $p$ vectors in $\mathbb{R}^n$ is linearly dependent whenever $p > n$ .
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