# Find a Spanning Set for the Vector Space of Skew-Symmetric Matrices

## Problem 714

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 $W$.

## Proof.

To prove that $W$ is a subspace of $V$, the $3\times 3$ zero matrix $\mathbf{0}$ is the zero vector of $V$, and since $\mathbf{0}$ is clearly skew-symmetric, $\mathbf{0}\in W$. Next, let $A,B\in W$. Then $A^{T}=-A$ and $B^{T}=-B$. Therefore,
$(A+B)^{T} =A^{T}+B^{T} =-A+(-B) =-(A+B).$ Thus $A+B$ is skew-symmetric, and therefore $A+B\in W$. Next, take any $A\in W$ and $r$ in the scalar field. Then
$(rA)^{T} =rA^{T} =r(-A) =-rA.$ Thus $rA$ is skew symmetric, which implies $rA\in W$. Therefore, $W$ is a subspace of $V$.

To exhibit a spanning set for $W$, first note that
$W= \left\{ A\left| A= \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix} ,\;a,b,c\in\R \right.\right\}.$ Let
$A_{1}= \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} ,\; A_{2}= \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{bmatrix} ,\; A_{3}= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{bmatrix} .$ Then any $A\in W$ can be written as
$A= \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix} =aA_{1}+bA_{2}+cA_{3},$ which is a linear combination of $A_{1},A_{2},A_{3}$. Thus $\{A_{1},A_{2},A_{3}\}$ is a spanning set for $W$.

### More from my site

• Linear Independent Vectors and the Vector Space Spanned By Them Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$. Let […]
• Does an Extra Vector Change the Span? 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 $\R^5$. 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 […]
• Can We Reduce the Number of Vectors in a Spanning Set? 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 $V$?   Solution. Yes, in general, $S_2$ can be a spanning set. As an […]
• Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient. Let $W$ be the following subset of $P_3$. $W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.$ Here $p'(x)$ is the first derivative of $p(x)$ and […]
• If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent Let $V$ be a subspace of $\R^n$. Suppose that $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}$ is a spanning set for $V$. Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.   We give two proofs. The essential ideas behind […]
• The Subspace of Linear Combinations whose Sums of Coefficients are zero Let $V$ be a vector space over a scalar field $K$. Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset \[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } […]
• The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$. Then prove that $V$ is a subspace of $\R^n$.   Proof. To prove that $V=\{\mathbf{0}\}$ is a subspace of $\R^n$, we check the following subspace […]
• Subspaces of Symmetric, Skew-Symmetric Matrices Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$. (a) The set $S$ consisting of all $n\times n$ symmetric matrices. (b) The set $T$ consisting of […]

#### You may also like...

##### Determine Bases for Nullspaces $\calN(A)$ and $\calN(A^{T}A)$

Determine bases for $\calN(A)$ and $\calN(A^{T}A)$ when \[ A= \begin{bmatrix} 1 & 2 & 1 \\ 1 & 1 &...

Close