20.5: Exercises

1

If $F$ is a field, show that $F[x]$ is a vector space over $F\text{,}$ where the vectors in $F[x]$ are polynomials. Vector addition is polynomial addition, and scalar multiplication is defined by $\alpha p(x)$ for $\alpha \in F\text{.}$

2

Prove that ${\mathbb Q }( \sqrt{2}\, )$ is a vector space.

3

Let ${\mathbb Q }( \sqrt{2}, \sqrt{3}\, )$ be the field generated by elements of the form $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}\text{,}$ where $a, b, c, d$ are in ${\mathbb Q}\text{.}$ Prove that ${\mathbb Q }( \sqrt{2}, \sqrt{3}\, )$ is a vector space of dimension $4$ over ${\mathbb Q}\text{.}$ Find a basis for ${\mathbb Q }( \sqrt{2}, \sqrt{3}\, )\text{.}$

4

Prove that the complex numbers are a vector space of dimension $2$ over ${\mathbb R}\text{.}$

5

Prove that the set $P_n$ of all polynomials of degree less than $n$ form a subspace of the vector space $F[x]\text{.}$ Find a basis for $P_n$ and compute the dimension of $P_n\text{.}$

6

Let $F$ be a field and denote the set of $n$-tuples of $F$ by $F^n\text{.}$ Given vectors $u = (u_1, \ldots, u_n)$ and $v = (v_1, \ldots, v_n)$ in $F^n$ and $\alpha$ in $F\text{,}$ define vector addition by

$$u + v = (u_1, \ldots, u_n) + (v_1, \ldots, v_n) = (u_1 + v_1, \ldots, u_n + v_n) \nonumber$$

and scalar multiplication by

$$\alpha u = \alpha(u_1, \ldots, u_n)= (\alpha u_1, \ldots, \alpha u_n)\text{.} \nonumber$$

Prove that $F^n$ is a vector space of dimension $n$ under these operations.

7

Which of the following sets are subspaces of ${\mathbb R}^3\text{?}$ If the set is indeed a subspace, find a basis for the subspace and compute its dimension.

1. $\displaystyle \{ (x_1, x_2, x_3) : 3 x_1 - 2 x_2 + x_3 = 0 \}$
2. $\displaystyle \{ (x_1, x_2, x_3) : 3 x_1 + 4 x_3 = 0, 2 x_1 - x_2 + x_3 = 0 \}$
3. $\displaystyle \{ (x_1, x_2, x_3) : x_1 - 2 x_2 + 2 x_3 = 2 \}$
4. $\displaystyle \{ (x_1, x_2, x_3) : 3 x_1 - 2 x_2^2 = 0 \}$

8

Show that the set of all possible solutions $(x, y, z) \in {\mathbb R}^3$ of the equations

$$\begin{align*} Ax + B y + C z & = 0\\ D x + E y + C z & = 0 \end{align*}$$

form a subspace of ${\mathbb R}^3\text{.}$

9

Let $W$ be the subset of continuous functions on $[0, 1]$ such that $f(0) = 0\text{.}$ Prove that $W$ is a subspace of $C[0, 1]\text{.}$

10

Let $V$ be a vector space over $F\text{.}$ Prove that $-(\alpha v) = (-\alpha)v = \alpha(-v)$ for all $\alpha \in F$ and all $v \in V\text{.}$

11

Let $V$ be a vector space of dimension $n\text{.}$ Prove each of the following statements.

1. If $S = \{v_1, \ldots, v_n \}$ is a set of linearly independent vectors for $V\text{,}$ then $S$ is a basis for $V\text{.}$
2. If $S = \{v_1, \ldots, v_n \}$ spans $V\text{,}$ then $S$ is a basis for $V\text{.}$
3. If $S = \{v_1, \ldots, v_k \}$ is a set of linearly independent vectors for $V$ with $k \lt n\text{,}$ then there exist vectors $v_{k + 1}, \ldots, v_n$ such that

   $$\{v_1, \ldots, v_k, v_{k + 1}, \ldots, v_n \} \nonumber$$

   is a basis for $V\text{.}$

12

Prove that any set of vectors containing ${\mathbf 0}$ is linearly dependent.

13

Let $V$ be a vector space. Show that $\{ {\mathbf 0} \}$ is a subspace of $V$ of dimension zero.

14

If a vector space $V$ is spanned by $n$ vectors, show that any set of $m$ vectors in $V$ must be linearly dependent for $m \gt n\text{.}$

15. Linear Transformations

Let $V$ and $W$ be vector spaces over a field $F\text{,}$ of dimensions $m$ and $n\text{,}$ respectively. If $T: V \rightarrow W$ is a map satisfying

$$\begin{align*} T( u+ v ) & = T(u ) + T(v)\\ T( \alpha v ) & = \alpha T(v) \end{align*}$$

for all $\alpha \in F$ and all $u, v \in V\text{,}$ then $T$ is called a linear transformation from $V$ into $W\text{.}$

1. Prove that the kernel of $T\text{,}$ $\ker(T) = \{ v \in V : T(v) = {\mathbf 0} \}\text{,}$ is a subspace of $V\text{.}$ The kernel of $T$ is sometimes called the null space of $T\text{.}$
2. Prove that the range or range space of $T\text{,}$ $R(V) = \{ w \in W : T(v) = w \text{ for some } v \in V \}\text{,}$ is a subspace of $W\text{.}$
3. Show that $T : V \rightarrow W$ is injective if and only if $\ker(T) = \{ \mathbf 0 \}\text{.}$
4. Let $\{ v_1, \ldots, v_k \}$ be a basis for the null space of $T\text{.}$ We can extend this basis to be a basis $\{ v_1, \ldots, v_k, v_{k + 1}, \ldots, v_m\}$ of $V\text{.}$ Why? Prove that $\{ T(v_{k + 1}), \ldots, T(v_m) \}$ is a basis for the range of $T\text{.}$ Conclude that the range of $T$ has dimension $m - k\text{.}$
5. Let $\dim V = \dim W\text{.}$ Show that a linear transformation $T : V \rightarrow W$ is injective if and only if it is surjective.

16

Let $V$ and $W$ be finite dimensional vector spaces of dimension $n$ over a field $F\text{.}$ Suppose that $T: V \rightarrow W$ is a vector space isomorphism. If $\{ v_1, \ldots, v_n \}$ is a basis of $V\text{,}$ show that $\{ T(v_1), \ldots, T(v_n) \}$ is a basis of $W\text{.}$ Conclude that any vector space over a field $F$ of dimension $n$ is isomorphic to $F^n\text{.}$

17. Direct Sums

Let $U$ and $V$ be subspaces of a vector space $W\text{.}$ The sum of $U$ and $V\text{,}$ denoted $U + V\text{,}$ is defined to be the set of all vectors of the form $u + v\text{,}$ where $u \in U$ and $v \in V\text{.}$

1. Prove that $U + V$ and $U \cap V$ are subspaces of $W\text{.}$
2. If $U + V = W$ and $U \cap V = {\mathbf 0}\text{,}$ then $W$ is said to be the direct sum. In this case, we write $W = U \oplus V\text{.}$ Show that every element $w \in W$ can be written uniquely as $w = u + v\text{,}$ where $u \in U$ and $v \in V\text{.}$
3. Let $U$ be a subspace of dimension $k$ of a vector space $W$ of dimension $n\text{.}$ Prove that there exists a subspace $V$ of dimension $n-k$ such that $W = U \oplus V\text{.}$ Is the subspace $V$ unique?
4. If $U$ and $V$ are arbitrary subspaces of a vector space $W\text{,}$ show that

   $$\dim( U + V) = \dim U + \dim V - \dim( U \cap V)\text{.} \nonumber$$

18. Dual Spaces

Let $V$ and $W$ be finite dimensional vector spaces over a field $F\text{.}$

1. Show that the set of all linear transformations from $V$ into $W\text{,}$ denoted by $\Hom(V, W)\text{,}$ is a vector space over $F\text{,}$ where we define vector addition as follows:

   $$\begin{align*} (S + T)(v) & = S(v) +T(v)\\ (\alpha S)(v) & = \alpha S(v)\text{,} \end{align*}$$

   where $S, T \in \Hom(V, W)\text{,}$ $\alpha \in F\text{,}$ and $v \in V\text{.}$

2. Let $V$ be an $F$-vector space. Define the dual space of $V$ to be $V^* = \Hom(V, F)\text{.}$ Elements in the dual space of $V$ are called linear functionals. Let $v_1, \ldots, v_n$ be an ordered basis for $V\text{.}$ If $v = \alpha_1 v_1 + \cdots + \alpha_n v_n$ is any vector in $V\text{,}$ define a linear functional $\phi_i : V \rightarrow F$ by $\phi_i (v) = \alpha_i\text{.}$ Show that the $\phi_i$'s form a basis for $V^*\text{.}$ This basis is called the dual basis of $v_1, \ldots, v_n$ (or simply the dual basis if the context makes the meaning clear).
3. Consider the basis $\{ (3, 1), (2, -2) \}$ for ${\mathbb R}^2\text{.}$ What is the dual basis for $({\mathbb R}^2)^*\text{?}$
4. Let $V$ be a vector space of dimension $n$ over a field $F$ and let $V^{* *}$ be the dual space of $V^*\text{.}$ Show that each element $v \in V$ gives rise to an element $\lambda_v$ in $V^{**}$ and that the map $v \mapsto \lambda_v$ is an isomorphism of $V$ with $V^{**}\text{.}$