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Section20.4Exercises

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 \(V^{\ast \ast}\).