Expand a Set of Matrices into a Basis for M

Math Problem Statement

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

Solution

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:

The two matrices in $S$ are linearly independent.
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:

\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.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Basis
Linear Independence
Matrix Spaces

Formulas

Theorems

Dimension Theorem
Linear Independence Criteria

Suitable Grade Level

University level

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