Exercises20.5Exercises

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

\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)\text{.} \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 $${\mathbb R}^3\text{?}$$ 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

\begin{equation*} \{v_1, \ldots, v_k, v_{k + 1}, \ldots, v_n \} \end{equation*}

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

\begin{equation*} \dim( U + V) = \dim U + \dim V - \dim( U \cap V)\text{.} \end{equation*}
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 $$V^{**}\text{.}$$