Subspace Of 2x2 Real Matrices: Is W A Subspace Of V?

Alright guys, let's dive into some linear algebra! We've got a fun problem here where we're looking at matrices and trying to figure out if certain sets of them form subspaces. So, let's break it down. We're given that $V$ is the set of all $2 \,\times 2$ real matrices. Think of it as the universe of all $2 \times 2$ matrices with real number entries. Now, we have a subset $W$, and our mission is to determine whether $W$ is a subspace of $V$. Remember, for $W$ to be a subspace, it needs to satisfy three key conditions:

1. The zero vector (or in this case, the zero matrix) must be in $W$.
2. $W$ must be closed under addition. In other words, if you add any two matrices in $W$, the result must also be in $W$.
3. $W$ must be closed under scalar multiplication. Meaning if you multiply any matrix in $W$ by a scalar (a real number), the result must also be in $W$.

Let's tackle each part of the problem step by step. Grab your favorite beverage, and let's get started!

a) $W$ = The Set of All $2 \times 2$ Lower Triangular Matrices

So, our first task is to figure out if the set of all $2 \times 2$ lower triangular matrices forms a subspace of $V$. A lower triangular matrix is basically a matrix where all the entries above the main diagonal are zero. A general $2 \times 2$ lower triangular matrix looks like this:

$\begin{bmatrix} a & 0 \\ b & c \end{bmatrix}$

where $a$, $b$, and $c$ are real numbers. Let's check the three conditions for a subspace:

1. Zero Matrix

Is the zero matrix in $W$? The zero matrix is:

$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

This is a lower triangular matrix because all entries above the main diagonal are zero. So, the first condition is satisfied.

2. Closure Under Addition

Is $W$ closed under addition? Let's take two arbitrary lower triangular matrices, say $A$ and $B$:

$A = \begin{bmatrix} a_1 & 0 \\ b_1 & c_1 \end{bmatrix}$, $B = \begin{bmatrix} a_2 & 0 \\ b_2 & c_2 \end{bmatrix}$

Now, let's add them:

$A + B = \begin{bmatrix} a_1 + a_2 & 0 \\ b_1 + b_2 & c_1 + c_2 \end{bmatrix}$

Notice that the resulting matrix is also a lower triangular matrix because the entry above the main diagonal is still zero. Since the sum of two lower triangular matrices is another lower triangular matrix, $W$ is closed under addition. The second condition is satisfied.

3. Closure Under Scalar Multiplication

Is $W$ closed under scalar multiplication? Let's take a lower triangular matrix $A$ and a scalar $k$ (a real number):

$A = \begin{bmatrix} a & 0 \\ b & c \end{bmatrix}$

Now, let's multiply $A$ by $k$:

$kA = \begin{bmatrix} ka & 0 \\ kb & kc \end{bmatrix}$

Again, the resulting matrix is a lower triangular matrix because the entry above the main diagonal is still zero. Since multiplying a lower triangular matrix by a scalar results in another lower triangular matrix, $W$ is closed under scalar multiplication. The third condition is satisfied.

Since $W$ satisfies all three conditions, we can confidently say that the set of all $2 \times 2$ lower triangular matrices is a subspace of $V$.

b) $W = \{ A \mid A = A^T \}$

Now, let's tackle the second part of the problem. Here, $W$ is the set of all $2 \times 2$ matrices $A$ such that $A = A^T$. In other words, $W$ is the set of all symmetric $2 \times 2$ matrices. A matrix is symmetric if it is equal to its transpose. Remember that the transpose of a matrix is obtained by swapping its rows and columns. A general $2 \times 2$ symmetric matrix looks like this:

$\begin{bmatrix} a & b \\ b & c \end{bmatrix}$

where $a$, $b$, and $c$ are real numbers. Notice that the entries off the main diagonal are equal. Let's check the three conditions for a subspace:

1. Zero Matrix

Is the zero matrix in $W$? The zero matrix is:

$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

The transpose of the zero matrix is itself, so the zero matrix is symmetric. Thus, the zero matrix is in $W$, and the first condition is satisfied.

2. Closure Under Addition

Is $W$ closed under addition? Let's take two arbitrary symmetric matrices, say $A$ and $B$:

$A = \begin{bmatrix} a_1 & b_1 \\ b_1 & c_1 \end{bmatrix}$, $B = \begin{bmatrix} a_2 & b_2 \\ b_2 & c_2 \end{bmatrix}$

Now, let's add them:

$A + B = \begin{bmatrix} a_1 + a_2 & b_1 + b_2 \\ b_1 + b_2 & c_1 + c_2 \end{bmatrix}$

To check if $A + B$ is symmetric, we need to see if it's equal to its transpose. The transpose of $A + B$ is:

$(A + B)^T = \begin{bmatrix} a_1 + a_2 & b_1 + b_2 \\ b_1 + b_2 & c_1 + c_2 \end{bmatrix}^T = \begin{bmatrix} a_1 + a_2 & b_1 + b_2 \\ b_1 + b_2 & c_1 + c_2 \end{bmatrix}$

Since $A + B = (A + B)^T$, the sum of two symmetric matrices is also symmetric. Therefore, $W$ is closed under addition, and the second condition is satisfied.

3. Closure Under Scalar Multiplication

Is $W$ closed under scalar multiplication? Let's take a symmetric matrix $A$ and a scalar $k$ (a real number):

$A = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$

Now, let's multiply $A$ by $k$:

$kA = \begin{bmatrix} ka & kb \\ kb & kc \end{bmatrix}$

To check if $kA$ is symmetric, we need to see if it's equal to its transpose. The transpose of $kA$ is:

$(kA)^T = \begin{bmatrix} ka & kb \\ kb & kc \end{bmatrix}^T = \begin{bmatrix} ka & kb \\ kb & kc \end{bmatrix}$

Since $kA = (kA)^T$, multiplying a symmetric matrix by a scalar results in another symmetric matrix. Therefore, $W$ is closed under scalar multiplication, and the third condition is satisfied.

Since $W$ satisfies all three conditions, we can conclude that the set of all $2 \times 2$ symmetric matrices is a subspace of $V$.

Conclusion

So, there you have it! Both the set of all $2 \times 2$ lower triangular matrices and the set of all $2 \times 2$ symmetric matrices are subspaces of the vector space $V$ of all $2 \times 2$ real matrices. We verified this by checking the three essential conditions: the presence of the zero matrix, closure under addition, and closure under scalar multiplication. Hope this breakdown helps you better understand subspaces! Keep exploring the fascinating world of linear algebra!