##### 1

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

##### 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 $a + b \sqrt{2} + c \sqrt{3}$, where $a, b, c$ are in ${\mathbb Q}$. Prove that ${\mathbb Q }( \sqrt{2}, \sqrt{3}\, )$ is a vector space of dimension 4 over ${\mathbb Q}$. Find a basis for ${\mathbb Q }( \sqrt{2}, \sqrt{3}\, )$.

##### 4

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

##### 5

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

##### 6

Let $F$ be a field and denote the set of $n$-tuples of $F$ by $F^n$. Given vectors $u = (u_1, \ldots, u_n)$ and $v = (v_1, \ldots, v_n)$ in $F^n$ and $\alpha$ in $F$, define vector addition by \begin{equation*}u + v = (u_1, \ldots, u_n) + (v_1, \ldots, v_n) = (u_1 + v_1, \ldots, u_n + v_n)\end{equation*} and scalar multiplication by \begin{equation*}\alpha u = \alpha(u_1, \ldots, u_n)= (\alpha u_1, \ldots, \alpha u_n).\end{equation*} Prove that $F^n$ is a vector space of dimension $n$ under these operations.

##### 7

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

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

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

3. $\{ (x_1, x_2, x_3) : x_1 - 2 x_2 + 2 x_3 = 2 \}$

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

##### 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 ${\mathbb R}^3$.

##### 9

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

##### 10

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

##### 11

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

1. If $S = \{v_1, \ldots, v_n \}$ is a set of linearly independent vectors for $V$, then $S$ is a basis for $V$.

2. If $S = \{v_1, \ldots, v_n \}$ spans $V$, then $S$ is a basis for $V$.

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

##### 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 $m \gt n$.

##### 15Linear Transformations

Let $V$ and $W$ be vector spaces over a field $F$, of dimensions $m$ and $n$, 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 $u, v \in V$, then $T$ is called a linear transformation from $V$ into $W$.

1. Prove that the kernel of $T$, $\ker(T) = \{ v \in V : T(v) = {\mathbf 0} \}$, is a subspace of $V$. The kernel of $T$ is sometimes called the null space of $T$.

2. Prove that the range or range space of $T$, $R(V) = \{ w \in W : T(v) = w \text{ for some } v \in V \}$, is a subspace of $W$.

3. Show that $T : V \rightarrow W$ is injective if and only if $\ker(T) = \{ \mathbf 0 \}$.

4. Let $\{ v_1, \ldots, v_k \}$ be a basis for the null space of $T$. We can extend this basis to be a basis $\{ v_1, \ldots, v_k, v_{k + 1}, \ldots, v_m\}$ of $V$. Why? Prove that $\{ T(v_{k + 1}), \ldots, T(v_m) \}$ is a basis for the range of $T$. Conclude that the range of $T$ has dimension $m-k$.

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

##### 17Direct Sums

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

1. Prove that $U + V$ and $U \cap V$ are subspaces of $W$.

2. If $U + V = W$ and $U \cap V = {\mathbf 0}$, then $W$ is said to be the direct sum. In this case, we write $W = U \oplus V$. Show that every element $w \in W$ can be written uniquely as $w = u + v$, where $u \in U$ and $v \in V$.

3. Let $U$ be a subspace of dimension $k$ of a vector space $W$ of dimension $n$. Prove that there exists a subspace $V$ of dimension $n-k$ such that $W = U \oplus V$. Is the subspace $V$ unique?

4. If $U$ and $V$ are arbitrary subspaces of a vector space $W$, show that \begin{equation*}\dim( U + V) = \dim U + \dim V - \dim( U \cap V).\end{equation*}

##### 18Dual Spaces

Let $V$ and $W$ be finite dimensional vector spaces over a field $F$.

1. Show that the set of all linear transformations from $V$ into $W$, denoted by $\Hom(V, W)$, is a vector space over $F$, where we define vector addition as follows: \begin{align*} (S + T)(v) & = S(v) +T(v)\\ (\alpha S)(v) & = \alpha S(v), \end{align*} where $S, T \in \Hom(V, W)$, $\alpha \in F$, and $v \in V$.

2. Let $V$ be an $F$-vector space. Define the dual space of $V$ to be $V^\ast = \Hom(V, F)$. Elements in the dual space of $V$ are called linear functionals. Let $v_1, \ldots, v_n$ be an ordered basis for $V$. If $v = \alpha_1 v_1 + \cdots + \alpha_n v_n$ is any vector in $V$, define a linear functional $\phi_i : V \rightarrow F$ by $\phi_i (v) = \alpha_i$. Show that the $\phi_i$'s form a basis for $V^\ast$. 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 ${\mathbb R}^2$. What is the dual basis for $({\mathbb R}^2)^\ast$?

4. Let $V$ be a vector space of dimension $n$ over a field $F$ and let $V^{\ast \ast}$ be the dual space $V^\ast$. Show that each element $v \in V$ gives rise to an element $\lambda_v$ in $V^{\ast \ast}$ and that the map $v \mapsto \lambda_v$ is an isomorphism of $V$ with $V^{\ast \ast}$.