Decomposing space of 2x2 matrices over the reals

You can then try to construct a basis for ##V## by considering the standard basis for ##\mathbb{M}^2(\mathbb{R})## and removing a suitable set of vectors.In summary, the conversation discusses the subspace W of M^2(R) and questions about decomposing it. The first question asks if there is a subspace V such that W and V together make up the entire space M^2(R). The second question asks if there is a different subspace with the same property. The conversation also considers the possibility of a different proper subspace U that, when added to W, would make up M^2(R). The attempt at a solution suggests finding the dimension of W and constructing
  • #1
Mathkid3242
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0

Homework Statement


Consider the subspace $$W:=\Bigl \{ \begin{bmatrix}
a & b \\
b & a \end{bmatrix} : a,b \in \mathbb{R}\Bigr \}$$ of $$\mathbb{M}^2(\mathbb{R}). $$

I have a few questions about how this can be decomposed.

1) Is there a subspace $$V$$ of $$\mathbb{M}^2(\mathbb{R})$$ such that $$W\oplus V=\mathbb{M}^2(\mathbb{R})$$? If so, what is one?

2) Further, is there a different (i.e., $$\ne V$$) subspace with the same property? And if not, is there a different proper subspace $$U$$ such that $$W+U= \mathbb{M}^2(\mathbb{R})$$?
Which examples, if any, would work for these questions? I haven't made much progress, so seeing explicit examples would help.

Homework Equations

The Attempt at a Solution



I tried considering this. Set $$V= \{A\in \mathbb{M}^2(\mathbb{R}): A^T =-A\}$$, but it didn't work.[/B]
 
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  • #2
A good start would be to find the dimension of ##W##. Then, you know that the dimension of ##V##, if it exists, should be ##4 - dim(W)##.
 
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1. What is the decomposing space of 2x2 matrices over the reals?

The decomposing space of 2x2 matrices over the reals refers to the set of all possible ways in which a 2x2 matrix can be broken down into simpler components. This can include diagonalization, triangularization, and other methods of simplification.

2. Why is the decomposing space of 2x2 matrices over the reals important?

The decomposing space of 2x2 matrices over the reals is important because it allows us to better understand and analyze the properties and behavior of matrices. By breaking them down into simpler components, we can more easily solve equations involving matrices and make predictions about their behavior.

3. How is the decomposing space of 2x2 matrices over the reals calculated?

The decomposing space of 2x2 matrices over the reals is typically calculated by using linear algebra techniques such as finding eigenvalues and eigenvectors, performing matrix operations such as row reduction, and using other methods such as the Cayley-Hamilton theorem.

4. Can the decomposing space of 2x2 matrices over the reals be applied to other dimensions?

Yes, the concept of decomposing matrices into simpler components can be applied to matrices of any dimension. However, the specific methods and techniques used may differ depending on the size and properties of the matrix.

5. What real-world applications can be found for the decomposing space of 2x2 matrices over the reals?

The decomposing space of 2x2 matrices over the reals has various applications in fields such as physics, engineering, and computer science. It can be used to solve systems of linear equations, model physical systems, and analyze data in machine learning algorithms, among many other uses.

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