Quiz 8. Determine Subsets are Subspaces: Functions Taking Integer Values / Set of Skew-Symmetric Matrices

Problem 328

(a) Let $C[-1,1]$ be the vector space over $\R$ of all real-valued continuous functions defined on the interval $[-1, 1]$.
Consider the subset $F$ of $C[-1, 1]$ defined by
\[F=\{ f(x)\in C[-1, 1] \mid f(0) \text{ is an integer}\}.\] Prove or disprove that $F$ is a subspace of $C[-1, 1]$.

(b) Let $n$ be a positive integer.
An $n\times n$ matrix $A$ is called skew-symmetric if $A^{\trans}=-A$.
Let $M_{n\times n}$ be the vector space over $\R$ of all $n\times n$ real matrices.
Consider the subset $W$ of $M_{n\times n}$ defined by
\[W=\{A\in M_{n\times n} \mid A \text{ is skew-symmetric}\}.\] Prove or disprove that $W$ is a subspace of $M_{n\times n}$.

 
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Proof.

(a) Continuous functions taking integer values at $0$.

We claim that the subset $F$ is not a subspace of the vector space $C[-1, 1]$.
If $F$ is a subspace, then $F$ is closed under scalar multiplication.
For instance, consider the scalar multiplication of the function $f(x)=x+1$ and the scalar $r=1/2$.
Note that since $f(0)=1$ is an integer, the function is in $F$.
However, the scalar multiplication
\[rf(x)=\frac{1}{2}(x+1)\] is not in $F$ because
\[rf(0)=\frac{1}{2}\] is not an integer.
Therefore, $F$ is not closed under scalar multiplication, hence $F$ is not a subspace of $C[-1, 1]$.

(b) Set of all $n\times n$ skew-symmetric matrices

We prove that $W$ is a subspace of $V$ by showing the following subspace criteria.

1. The zero vector $\mathbf{0}\in M_{n\times n}$ is in $W$.
2. If $A, B\in W$, then $A+B\in W$.
3. If $A\in W, r\in \R$, then $rA\in W$.

First, note that the zero vector in the vector space $M_{n\times n}$ is the $n\times n$ zero matrix $O$. Since we have
\[O^{\trans}=O=-O,\] the zero matrix $O$ is skew-symmetric, hence the zero vector $O$ is in $W$. So condition 1 is met.

To prove condition 2, let $A, B$ be arbitrary elements in $W$. Thus, $A$ and $B$ are skew-symmetric matrices:
\[A^{\trans}=-A \text{ and } B^{\trans}=-B. \tag{*}\] We want to prove that the addition $A+B$ is in $W$, namely, we want to show $A+B$ is skew-symmetric.

We have
\begin{align*}
(A+B)^{\trans}=A^{\trans}+B^{\trans}\stackrel{(*)}{=}-A+(-B)=-(A+B),
\end{align*}
and this implies that $A+B$ is skew-symmetric. Hence $A+B \in W$ and condition 2 is satisfied.

Finally, we check condition 3. Let $A\in W$ and $r\in \R$. We want to show that the scalar product $rA\in W$, that is, $rA$ is skew-symmetric.
We have
\begin{align*}
(rA)^{\trans} &= rA^{\trans} \\
&= r(-A) \qquad \qquad (\text{$A$ is skew-symmetric})\\
&=-rA,
\end{align*}
which yields that $rA$ is skew-symmetric. Hence $rA \in W$ and condition 3 is met as well.

Thus, we have checked all the subspace criteria. Hence $W$ is a subspace of $M_{n\times n}$.

Comment.

These are Quiz 8 problems for Math 2568 (Introduction to Linear Algebra) at OSU in Spring 2017.

List of Quiz Problems of Linear Algebra (Math 2568) at OSU in Spring 2017

There were 13 weekly quizzes. Here is the list of links to the quiz problems and solutions.

• Quiz 1. Gauss-Jordan elimination / homogeneous system.
• Quiz 2. The vector form for the general solution / Transpose matrices.
• Quiz 3. Condition that vectors are linearly dependent/ orthogonal vectors are linearly independent
• Quiz 4. Inverse matrix/ Nonsingular matrix satisfying a relation
• Quiz 5. Example and non-example of subspaces in 3-dimensional space
• Quiz 6. Determine vectors in null space, range / Find a basis of null space
• Quiz 7. Find a basis of the range, rank, and nullity of a matrix
• Quiz 8. Determine subsets are subspaces: functions taking integer values / set of skew-symmetric matrices
• Quiz 9. Find a basis of the subspace spanned by four matrices
• Quiz 10. Find orthogonal basis / Find value of linear transformation
• Quiz 11. Find eigenvalues and eigenvectors/ Properties of determinants
• Quiz 12. Find eigenvalues and their algebraic and geometric multiplicities
• Quiz 13 (Part 1). Diagonalize a matrix.
• Quiz 13 (Part 2). Find eigenvalues and eigenvectors of a special matrix

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