Math Problems Vault

Problem 2B.1

Find all vector spaces that have exactly one basis.

?

Problem 2B.2

Verify all assertions in Example 2.27.

Example 2.27:

?

(a)

The list $(1,0,\ldots ,0), (0,1,0,\ldots ,0), \ldots , (0,\ldots ,0,1)$ is a basis of $\mathbf{F}^{n}$ , called the standard basis of $\mathbf{F}^{n}$ .

(b)

The list $(1,2), (3,5)$ is a basis of $\mathbf{F}^{2}$ . Note that this list has length two, which is the same as the length of the standard basis of $\mathbf{F}^{2}$ .

(c)

The list $(1,2,-4), (7,-5,6)$ is linearly independent in $\mathbf{F}^{3}$ but is not a basis of $\mathbf{F}^{3}$ because it does not span $\mathbf{F}^{3}$ .

(d)

The list $(1,2), (3,5), (4,13)$ spans $\mathbf{F}^{2}$ but is not a basis of $\mathbf{F}^{2}$ because it is not linearly independent.

(e)

The list $(1,1,0), (0,0,1)$ is a basis of $\left\{ (x,x,y) \in \mathbf{F}^{3} : x, y \in \mathbf{F}\right\}$ .

(f)

The list $(1,-1,0), (1,0,-1)$ is a basis of $\left\{ (x,y,z) \in \mathbf{F}^{3} : x+y+z = 0\right\}$ .

(g)

The list $1, z, \ldots , z^{m}$ is a basis of $\mathcal{P}_{m}(\mathbf{F})$ , called the standard basis of $\mathcal{P}_{m}(\mathbf{F})$ .

Problem 2B.3

(a)(b)(c)

?

(a)

Let $U$ be the subspace of $\mathbb {R}^{5}$ defined by

U = \left\{ \left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \in \mathbb {R}^{5}: x_{1} = 3 x_{2} \text{ and } x_{3} = 7 x_{4}\right\} .

Find a basis of $U$ .

(b)

Extend the basis in (a) to a basis of $\mathbb {R}^{5}$ .

(c)

Find a subspace $W$ of $\mathbb {R}^{5}$ such that $\mathbb {R}^{5} = U \oplus W$ .

Problem 2B.4

(a)(b)(c)

?

(a)

Let $U$ be the subspace of $\mathbb {C}^{5}$ defined by

U = \left\{ \left(z_{1}, z_{2}, z_{3}, z_{4}, z_{5}\right) \in \mathbb {C}^{5}: 6 z_{1} = z_{2} \text{ and } z_{3}+2 z_{4}+3 z_{5} = 0\right\}

Find a basis of $U$ .

(b)

Extend the basis in (a) to a basis of $\mathbb {C}^{5}$ .

(c)

Find a subspace $W$ of $\mathbb {C}^{5}$ such that $\mathbb {C}^{5} = U \oplus W$ .

Problem 2B.5

Suppose $V$ is finite-dimensional and $U, W$ are subspaces of $V$ such that $V = U+W$ . Prove that there exists a basis of $V$ consisting of vectors in $U \cup W$ .

?

Problem 2B.6

Prove or give a counterexample: If $p_{0}, p_{1}, p_{2}, p_{3}$ is a list in $\mathcal{P}_{3}(\mathbf{F})$ such that none of the polynomials $p_{0}, p_{1}, p_{2}, p_{3}$ has degree 2, then $p_{0}, p_{1}, p_{2}, p_{3}$ is not a basis of $\mathcal{P}_{3}(\mathbf{F})$ .

?

Problem 2B.7

Suppose $v_{1}, v_{2}, v_{3}, v_{4}$ is a basis of $V$ . Prove that

v_{1}+v_{2}, v_{2}+v_{3}, v_{3}+v_{4}, v_{4}

is also a basis of $V$ .

?

Problem 2B.8

Prove or give a counterexample: If $v_{1}, v_{2}, v_{3}, v_{4}$ is a basis of $V$ and $U$ is a subspace of $V$ such that $v_{1}, v_{2} \in U$ and $v_{3} \notin U$ and $v_{4} \notin U$ , then $v_{1}, v_{2}$ is a basis of $U$ .

?

Problem 2B.9

Suppose $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$ . For $k \in \left\{ 1, \ldots , m\right\}$ , let

w_{k} = v_{1}+\cdots +v_{k}.

Show that $v_{1}, \ldots , v_{m}$ is a basis of $V$ if and only if $w_{1}, \ldots , w_{m}$ is a basis of $V$ .

?

Problem 2B.10

Suppose $U$ and $W$ are subspaces of $V$ such that $V = U \oplus W$ . Suppose also that $u_{1}, \ldots , u_{m}$ is a basis of $U$ and $w_{1}, \ldots , w_{n}$ is a basis of $W$ . Prove that

u_{1}, \ldots , u_{m}, w_{1}, \ldots , w_{n}

is a basis of $V$ .

?

Problem 2B.11

Suppose $V$ is a real vector space. Show that if $v_{1}, \ldots , v_{n}$ is a basis of $V$ (as a real vector space), then $v_{1}, \ldots , v_{n}$ is also a basis of the complexification $V_{\mathrm{C}}$ (as a complex vector space).

?

Remark.

?

The complexification $V_{\mathrm{C}}$ was defined in Exercise 8 in Section 1B: suppose $V$ is a real vector space; the complexification of $V$ , denoted by $V_{\mathrm{C}}$ , equals $V \times V$ , with an element $(u,v)$ , $u, v \in V$ , written as $u+iv$ ; addition on $V_{\mathrm{C}}$ is defined by $(u_1+iv_1)+(u_2+iv_2) = (u_1+u_2)+i(v_1+v_2)$ ; complex scalar multiplication on $V_{\mathrm{C}}$ is defined by $(a+bi)(u+iv) = (au-bv)+i(av+bu)$ for all $a, b \in \mathbb {R}$ and all $u, v \in V$ .