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The problem asks to expand the given set of matrices into a basis for \( M_{2 \times 2} \), the space of all \( 2 \times 2 \) matrices. The matrix space \( M_{2 \times 2} \) has dimension 4, meaning that we need 4 linearly independent matrices to form a basis.

Given the set:

\[
S = \left\{ \begin{pmatrix} 1 & 2 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \right\}
\]

We currently have 2 matrices. To form a complete basis, we need 2 more linearly independent matrices.

### Steps:
1. The two matrices in \( S \) are linearly independent.
2. To create a basis, we need to find 2 additional matrices such that the total set of matrices remains linearly independent and spans the entire space.

A typical basis for \( M_{2 \times 2} \) is the standard basis, which consists of the following 4 matrices:

\[
B = \left\{
\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, 
\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, 
\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, 
\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
\right\}
\]

Now, we can check which of these matrices are linearly independent from the matrices in \( S \), and select two of them to complete the basis.

### Expanded Basis:

1. The matrix \( \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \) is independent of the two given matrices, so we can add it to the basis.
2. Similarly, the matrix \( \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \) is also independent of the given set and can be added.

Thus, the expanded basis for \( M_{2 \times 2} \) becomes:

\[
\left\{ \begin{pmatrix} 1 & 2 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \right\}
\]

This forms a complete basis for \( M_{2 \times 2} \).

Would you like further details or explanations?

Here are five related questions for deeper understanding:

1. What is the general process for determining linear independence of matrices?
2. How can you verify if a set of matrices forms a basis for \( M_{2 \times 2} \)?
3. Why does \( M_{2 \times 2} \) have a dimension of 4?
4. What is the difference between the standard basis and other possible bases for \( M_{2 \times 2} \)?
5. Can you apply a similar method to find a basis for higher-dimensional matrix spaces?

**Tip**: When expanding a set of vectors or matrices into a basis, always ensure the resulting set is linearly independent and spans the entire space.

Perluas himpunan matriks berikut sehingga menjadi basis M_{22}, s = {(1 2 0, 1 0), (0 0 0 1)}

Learn how to expand a given set of matrices to form a basis for the matrix space M_{2×2}. This problem involves key linear algebra concepts such as matrix spaces, linear independence, and the dimension theorem. Suitable for university-level students.

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