Question 1166345: Yes, **W**, the set of skew-symmetric $n \times n$ matrices, is a **subspace** of $\mathbf{V}$, the vector space of all $n \times n$ matrices.
To prove that $W$ is a subspace of $V$, we must verify the three conditions for a subset to be a subspace:
1. **The Zero Vector Condition:** The zero vector of $V$ must be in $W$.
2. **Closure under Vector Addition:** If $\mathbf{A}$ and $\mathbf{B}$ are in $W$, then their sum $(\mathbf{A} + \mathbf{B})$ must also be in $W$.
3. **Closure under Scalar Multiplication:** If $\mathbf{A}$ is in $W$ and $c$ is any scalar, then the scalar multiple $(c\mathbf{A})$ must also be in $W$.
Recall that a matrix $\mathbf{A}$ is **skew-symmetric** if and only if $\mathbf{A}^T = -\mathbf{A}$.
***
### 1. The Zero Vector Condition
The zero vector in $V$ is the $n \times n$ **zero matrix**, $\mathbf{Z}$, where every entry is $0$.
We check if $\mathbf{Z}$ is skew-symmetric:
$$\mathbf{Z}^T = \mathbf{Z}$$
$$-\mathbf{Z} = -\mathbf{Z} \text{ (which is still } \mathbf{Z})$$
Since $\mathbf{Z}^T = \mathbf{Z} = -\mathbf{Z}$, the condition $\mathbf{Z}^T = -\mathbf{Z}$ holds.
Thus, the zero matrix $\mathbf{Z}$ is skew-symmetric, and $\mathbf{Z} \in W$.
### 2. Closure under Vector Addition
Let $\mathbf{A}$ and $\mathbf{B}$ be two arbitrary matrices in $W$. This means:
* $\mathbf{A}^T = -\mathbf{A}$
* $\mathbf{B}^T = -\mathbf{B}$
We need to check if their sum $(\mathbf{A} + \mathbf{B})$ is also skew-symmetric. We check the transpose of the sum:
$$(\mathbf{A} + \mathbf{B})^T = \mathbf{A}^T + \mathbf{B}^T \quad (\text{Property of Transpose})$$
Substitute the skew-symmetric conditions:
$$(\mathbf{A} + \mathbf{B})^T = (-\mathbf{A}) + (-\mathbf{B})$$
$$(\mathbf{A} + \mathbf{B})^T = -(\mathbf{A} + \mathbf{B})$$
Since the transpose of $(\mathbf{A} + \mathbf{B})$ is equal to $-(\mathbf{A} + \mathbf{B})$, the sum $(\mathbf{A} + \mathbf{B})$ is skew-symmetric.
Thus, $W$ is closed under matrix addition.
### 3. Closure under Scalar Multiplication
Let $\mathbf{A}$ be an arbitrary matrix in $W$ (so $\mathbf{A}^T = -\mathbf{A}$), and let $c$ be an arbitrary scalar.
We need to check if the scalar multiple $(c\mathbf{A})$ is also skew-symmetric. We check the transpose of the multiple:
$$(c\mathbf{A})^T = c(\mathbf{A}^T) \quad (\text{Property of Transpose})$$
Substitute the skew-symmetric condition for $\mathbf{A}$:
$$(c\mathbf{A})^T = c(-\mathbf{A})$$
$$(c\mathbf{A})^T = -(c\mathbf{A})$$
Since the transpose of $(c\mathbf{A})$ is equal to $-(c\mathbf{A})$, the scalar multiple $(c\mathbf{A})$ is skew-symmetric.
Thus, $W$ is closed under scalar multiplication.
***
Since $W$ satisfies all three conditions, it is a **subspace** of the vector space $V$.
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Would you like to know the **dimension** of the subspace $W$ of skew-symmetric $n \times n$ matrices?
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! Yes, **W**, the set of skew-symmetric $n \times n$ matrices, is a **subspace** of $\mathbf{V}$, the vector space of all $n \times n$ matrices.
To prove that $W$ is a subspace of $V$, we must verify the three conditions for a subset to be a subspace:
1. **The Zero Vector Condition:** The zero vector of $V$ must be in $W$.
2. **Closure under Vector Addition:** If $\mathbf{A}$ and $\mathbf{B}$ are in $W$, then their sum $(\mathbf{A} + \mathbf{B})$ must also be in $W$.
3. **Closure under Scalar Multiplication:** If $\mathbf{A}$ is in $W$ and $c$ is any scalar, then the scalar multiple $(c\mathbf{A})$ must also be in $W$.
Recall that a matrix $\mathbf{A}$ is **skew-symmetric** if and only if $\mathbf{A}^T = -\mathbf{A}$.
***
### 1. The Zero Vector Condition
The zero vector in $V$ is the $n \times n$ **zero matrix**, $\mathbf{Z}$, where every entry is $0$.
We check if $\mathbf{Z}$ is skew-symmetric:
$$\mathbf{Z}^T = \mathbf{Z}$$
$$-\mathbf{Z} = -\mathbf{Z} \text{ (which is still } \mathbf{Z})$$
Since $\mathbf{Z}^T = \mathbf{Z} = -\mathbf{Z}$, the condition $\mathbf{Z}^T = -\mathbf{Z}$ holds.
Thus, the zero matrix $\mathbf{Z}$ is skew-symmetric, and $\mathbf{Z} \in W$.
### 2. Closure under Vector Addition
Let $\mathbf{A}$ and $\mathbf{B}$ be two arbitrary matrices in $W$. This means:
* $\mathbf{A}^T = -\mathbf{A}$
* $\mathbf{B}^T = -\mathbf{B}$
We need to check if their sum $(\mathbf{A} + \mathbf{B})$ is also skew-symmetric. We check the transpose of the sum:
$$(\mathbf{A} + \mathbf{B})^T = \mathbf{A}^T + \mathbf{B}^T \quad (\text{Property of Transpose})$$
Substitute the skew-symmetric conditions:
$$(\mathbf{A} + \mathbf{B})^T = (-\mathbf{A}) + (-\mathbf{B})$$
$$(\mathbf{A} + \mathbf{B})^T = -(\mathbf{A} + \mathbf{B})$$
Since the transpose of $(\mathbf{A} + \mathbf{B})$ is equal to $-(\mathbf{A} + \mathbf{B})$, the sum $(\mathbf{A} + \mathbf{B})$ is skew-symmetric.
Thus, $W$ is closed under matrix addition.
### 3. Closure under Scalar Multiplication
Let $\mathbf{A}$ be an arbitrary matrix in $W$ (so $\mathbf{A}^T = -\mathbf{A}$), and let $c$ be an arbitrary scalar.
We need to check if the scalar multiple $(c\mathbf{A})$ is also skew-symmetric. We check the transpose of the multiple:
$$(c\mathbf{A})^T = c(\mathbf{A}^T) \quad (\text{Property of Transpose})$$
Substitute the skew-symmetric condition for $\mathbf{A}$:
$$(c\mathbf{A})^T = c(-\mathbf{A})$$
$$(c\mathbf{A})^T = -(c\mathbf{A})$$
Since the transpose of $(c\mathbf{A})$ is equal to $-(c\mathbf{A})$, the scalar multiple $(c\mathbf{A})$ is skew-symmetric.
Thus, $W$ is closed under scalar multiplication.
***
Since $W$ satisfies all three conditions, it is a **subspace** of the vector space $V$.
---
Would you like to know the **dimension** of the subspace $W$ of skew-symmetric $n \times n$ matrices?
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