Question Determine if the set of all symmetric $2 \times 2$ matrices, $W$ , is a subspace of $R^{2 \times 2}$ .

Studdy Solution

STEP 1

Assumptions
1. We are considering $2 \times 2$ matrices.
2. A symmetric matrix is one where the matrix is equal to its transpose.
3. $W$ is the set of all symmetric $2 \times 2$ matrices.
4. We are considering $W$ as a subset of the vector space of all $2 \times 2$ matrices over the real numbers, denoted by $\mathbb{R}^{2 \times 2}$ .
5. To prove that $W$ is a subspace of $\mathbb{R}^{2 \times 2}$ , we must show that it is closed under addition and scalar multiplication, and that it contains the zero vector.

STEP 2

Define a general symmetric $2 \times 2$ matrix. A symmetric matrix $A$ will have the form:
$A = \begin{bmatrix} a & b \\ b & c \\ \end{bmatrix}$
where $a, b, c \in \mathbb{R}$ .

STEP 3

Define the zero matrix in $\mathbb{R}^{2 \times 2}$ , which is also a symmetric matrix:
$0 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix}$

STEP 4

Show that the zero matrix is in $W$ :
Since the zero matrix is symmetric, it belongs to the set $W$ .

STEP 5

Define another symmetric $2 \times 2$ matrix $B$ :
$B = \begin{bmatrix} d & e \\ e & f \\ \end{bmatrix}$
where $d, e, f \in \mathbb{R}$ .

STEP 6

Show that the sum of two symmetric matrices $A$ and $B$ is also symmetric:
$A + B = \begin{bmatrix} a & b \\ b & c \\ \end{bmatrix} + \begin{bmatrix} d & e \\ e & f \\ \end{bmatrix} = \begin{bmatrix} a+d & b+e \\ b+e & c+f \\ \end{bmatrix}$

STEP 7

Check if the transpose of $A + B$ is equal to $A + B$ :
$(A + B)^T = \begin{bmatrix} a+d & b+e \\ b+e & c+f \\ \end{bmatrix}^T = \begin{bmatrix} a+d & b+e \\ b+e & c+f \\ \end{bmatrix} = A + B$

STEP 8

Since the transpose of $A + B$ is equal to $A + B$ , the sum is symmetric and therefore belongs to $W$ .

STEP 9

Let $k$ be a scalar in $\mathbb{R}$ . Show that the scalar multiple of a symmetric matrix $A$ is also symmetric:
$kA = k\begin{bmatrix} a & b \\ b & c \\ \end{bmatrix} = \begin{bmatrix} ka & kb \\ kb & kc \\ \end{bmatrix}$

STEP 10

Check if the transpose of $kA$ is equal to $kA$ :
$(kA)^T = \begin{bmatrix} ka & kb \\ kb & kc \\ \end{bmatrix}^T = \begin{bmatrix} ka & kb \\ kb & kc \\ \end{bmatrix} = kA$

STEP 11

Since the transpose of $kA$ is equal to $kA$ , the scalar multiple is symmetric and therefore belongs to $W$ .

STEP 12

Since $W$ contains the zero matrix, is closed under addition and scalar multiplication, and every element of $W$ is a symmetric matrix, $W$ satisfies all the criteria to be a subspace of $\mathbb{R}^{2 \times 2}$ .
The statement is True.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms