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 \(V^{**}\text{.}\)