4.5 Exercises

20 points · 4 extra credit

The Dimension of a Vector Space

1. 1. (2 pts) State the dimension of each of the following vector spaces, and give a basis in each case: $\mathbb{R}^5$, $P_3$, and $M_{2\times 2}$.

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2. 2. (3 pts) Find a basis and the dimension of the subspace $H = \{(x_1, x_2, x_3, x_4) : x_1 = 3x_2 \text{ and } x_3 = -x_4\}$ of $\mathbb{R}^4$.

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3. 3. (3 pts) Find a basis and the dimension of the subspace of $P_2$ consisting of all polynomials $p$ with $p'(0) = 0$.

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4. 4. (2 pts) Find the dimension of the subspace $H = \text{Span}\,\{(1,2,1),\,(2,4,2),\,(3,6,3)\}$ of $\mathbb{R}^3$.

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5. 5. (3 pts) Suppose $\dim V = 4$. State whether each of the following is true or false, and justify each answer.

   (a) Any set of $5$ vectors in $V$ is linearly dependent.

   (b) Any set of $3$ vectors in $V$ spans $V$.

   (c) Any linearly independent set of exactly $4$ vectors in $V$ is a basis for $V$.

   (d) Any spanning set for $V$ must contain at least $4$ vectors.

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6. 6. (3 pts) Let $H$ be a subspace of a finite-dimensional vector space $V$. Explain why $\dim H \le \dim V$, and explain why $\dim H = \dim V$ forces $H = V$.

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7. 7. (2 pts) Find the dimension of the subspace of $M_{2\times 2}$ consisting of all symmetric $2\times 2$ matrices, and give a basis.

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8. 8. (2 pts) Find the dimension of the solution space of the homogeneous system whose coefficient matrix is $\begin{bmatrix} 1 & -2 & 0 & 3 \\ 0 & 0 & 1 & -1 \end{bmatrix}$.

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9. 9. (4 pts ••) Prove that in an $n$-dimensional vector space $V$, every linearly independent set contains at most $n$ vectors.

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