## The Number of Elements in a Finite Field is a Power of a Prime Number

## Problem 726

Let $\F$ be a finite field of characteristic $p$.

Prove that the number of elements of $\F$ is $p^n$ for some positive integer $n$.

Add to solve laterVector Space Problems and Solutions.

The other popular topics in Linear Algebra are

Gauss-Jordan Elimination Inverse Matrix Eigen Value Caley-Hamilton Theorem Caley-Hamilton Theorem

Check out the list of all problems in Linear Algebra

Let $\F$ be a finite field of characteristic $p$.

Prove that the number of elements of $\F$ is $p^n$ for some positive integer $n$.

Add to solve later Define two functions $T:\R^{2}\to\R^{2}$ and $S:\R^{2}\to\R^{2}$ by

\[

T\left(

\begin{bmatrix}

x \\ y

\end{bmatrix}

\right)

=

\begin{bmatrix}

2x+y \\ 0

\end{bmatrix}

,\;

S\left(

\begin{bmatrix}

x \\ y

\end{bmatrix}

\right)

=

\begin{bmatrix}

x+y \\ xy

\end{bmatrix}

.

\]
Determine whether $T$, $S$, and the composite $S\circ T$ are linear transformations.

Let $W$ be the set of $3\times 3$ skew-symmetric matrices. Show that $W$ is a subspace of the vector space $V$ of all $3\times 3$ matrices. Then, exhibit a spanning set for $W$.

Add to solve later Determine bases for $\calN(A)$ and $\calN(A^{T}A)$ when

\[

A=

\begin{bmatrix}

1 & 2 & 1 \\

1 & 1 & 3 \\

0 & 0 & 0

\end{bmatrix}

.

\]
Then, determine the ranks and nullities of the matrices $A$ and $A^{\trans}A$.

Using the axiom of a vector space, prove the following properties.

Let $V$ be a vector space over $\R$. Let $u, v, w\in V$.

**(a)** If $u+v=u+w$, then $v=w$.

**(b)** If $v+u=w+u$, then $v=w$.

**(c)** The zero vector $\mathbf{0}$ is unique.

**(d)** For each $v\in V$, the additive inverse $-v$ is unique.

**(e)** $0v=\mathbf{0}$ for every $v\in V$, where $0\in\R$ is the zero scalar.

**(f)** $a\mathbf{0}=\mathbf{0}$ for every scalar $a$.

**(g)** If $av=\mathbf{0}$, then $a=0$ or $v=\mathbf{0}$.

**(h)** $(-1)v=-v$.

The first two properties are called the **cancellation law**.

Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where

\[

\mathbf{v}_{1}=

\begin{bmatrix}

1 \\ 2 \\ 2 \\ -1

\end{bmatrix}

,\;\mathbf{v}_{2}=

\begin{bmatrix}

1 \\ 3 \\ 1 \\ 1

\end{bmatrix}

,\;\mathbf{v}_{3}=

\begin{bmatrix}

1 \\ 5 \\ -1 \\ 5

\end{bmatrix}

,\;\mathbf{v}_{4}=

\begin{bmatrix}

1 \\ 1 \\ 4 \\ -1

\end{bmatrix}

,\;\mathbf{v}_{5}=

\begin{bmatrix}

2 \\ 7 \\ 0 \\ 2

\end{bmatrix}

.\]
Find a basis for the span $\Span(S)$.

Let $A=\begin{bmatrix}

2 & 4 & 6 & 8 \\

1 &3 & 0 & 5 \\

1 & 1 & 6 & 3

\end{bmatrix}$.

**(a)** Find a basis for the nullspace of $A$.

**(b)** Find a basis for the row space of $A$.

**(c)** Find a basis for the range of $A$ that consists of column vectors of $A$.

**(d)** For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.

Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^3$. Is it possible that $S_2=\{\mathbf{v}_1\}$ is a spanning set for $V$?

Add to solve laterSuppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set

\[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\]
still a spanning set for $V$? If so, prove it. Otherwise, give a counterexample.

For a set $S$ and a vector space $V$ over a scalar field $\K$, define the set of all functions from $S$ to $V$

\[ \Fun ( S , V ) = \{ f : S \rightarrow V \} . \]

For $f, g \in \Fun(S, V)$, $z \in \K$, addition and scalar multiplication can be defined by

\[ (f+g)(s) = f(s) + g(s) \, \mbox{ and } (cf)(s) = c (f(s)) \, \mbox{ for all } s \in S . \]

**(a)** Prove that $\Fun(S, V)$ is a vector space over $\K$. What is the zero element?

**(b)** Let $S_1 = \{ s \}$ be a set consisting of one element. Find an isomorphism between $\Fun(S_1 , V)$ and $V$ itself. Prove that the map you find is actually a linear isomorpism.

**(c)** Suppose that $B = \{ e_1 , e_2 , \cdots , e_n \}$ is a basis of $V$. Use $B$ to construct a basis of $\Fun(S_1 , V)$.

**(d)** Let $S = \{ s_1 , s_2 , \cdots , s_m \}$. Construct a linear isomorphism between $\Fun(S, V)$ and the vector space of $n$-tuples of $V$, defined as

\[ V^m = \{ (v_1 , v_2 , \cdots , v_m ) \mid v_i \in V \mbox{ for all } 1 \leq i \leq m \} . \]

**(e)** Use the basis $B$ of $V$ to constract a basis of $\Fun(S, V)$ for an arbitrary finite set $S$. What is the dimension of $\Fun(S, V)$?

**(f)** Let $W \subseteq V$ be a subspace. Prove that $\Fun(S, W)$ is a subspace of $\Fun(S, V)$.

Let $A=\begin{bmatrix}

2 & 4 & 6 & 8 \\

1 &3 & 0 & 5 \\

1 & 1 & 6 & 3

\end{bmatrix}$.

**(a)** Find a basis for the nullspace of $A$.

**(b)** Find a basis for the row space of $A$.

**(c)** Find a basis for the range of $A$ that consists of column vectors of $A$.

**(d)** For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.

Using the definition of the range of a matrix, describe the range of the matrix

\[A=\begin{bmatrix}

2 & 4 & 1 & -5 \\

1 &2 & 1 & -2 \\

1 & 2 & 0 & -3

\end{bmatrix}.\]

Let $A=\begin{bmatrix}

1 & 0 & 3 & -2 \\

0 &3 & 1 & 1 \\

1 & 3 & 4 & -1

\end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$.

**(a)** $\begin{bmatrix}

-3 \\

0 \\

1 \\

0

\end{bmatrix}$

**(b)** $\begin{bmatrix}

-4 \\

-1 \\

2 \\

1

\end{bmatrix}$

**(c) **$\begin{bmatrix}

0 \\

0 \\

0 \\

0

\end{bmatrix}$

**(d)** $\begin{bmatrix}

0 \\

0 \\

0

\end{bmatrix}$

Then, describe the nullspace $\calN(A)$ of the matrix $A$.

Add to solve laterLet $V$ denote the vector space of $2 \times 2$ matrices, and $W$ the vector space of $3 \times 2$ matrices. Define the linear transformation $T : V \rightarrow W$ by

\[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = \begin{bmatrix} a+b & 2d \\ 2b – d & -3c \\ 2b – c & -3a \end{bmatrix}.\]

Find a basis for the range of $T$.

Add to solve laterLet $V$ be the vector space of $k \times k$ matrices. Then for fixed matrices $R, S \in V$, define the subset $W = \{ R A S \mid A \in V \}$.

Prove that $W$ is a vector subspace of $V$.

Add to solve laterLet $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by

\[ \ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .\]

Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = 0$.

Add to solve laterFor what real values of $a$ is the set

\[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\]
a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?

Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define

\[W = \{ A \in V \mid AM = MA \}.\]
The set $W$ here is called the **centralizer** of $M$ in $V$.

Prove that $W$ is a subspace of $V$.

Add to solve laterLet $V$ be the vector space of $n \times n$ matrices with real coefficients, and define

\[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\]
The set $W$ is called the **center** of $V$.

Prove that $W$ is a subspace of $V$.

Add to solve laterLet $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.

Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.

**(a)** Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$.

**(b)** Prove that the set $\{\sin^2(x)-\cos^2(x), 1\}$ is a basis for $W$.