Is the Intersection of Two Matrix Subspaces Nontrivial?

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In summary, The subspace U is defined as the set of all 2x2 matrices with the form (x, -x; y, z). The question is whether this set can be expressed as the linear combination of three matrices, M1, M2, and M3. By showing that these three matrices are independent, it can be concluded that they form a basis for U, with a dimension of 3. Another question is posed about the set W, which can be expressed as the linear combination of three matrices. The basis for this set is then given as three matrices with specific values. Finally, a question is asked about the intersection of these two sets.
  • #1
gazzo
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Hey, I have a quick question. :confused:

Let [itex]{\cal U}[/itex] be the subspace of [itex]\mathbb{R}_{2x2}[/itex] of all matrices of the form [tex]\left( \begin{array}{ccc}x&-x\\y&z\end{array}\right)[/tex].

Is it true, that

[tex]\left( \begin{array}{ccc}x&-x\\y&z\end{array}\right) = x\left( \begin{array}{ccc}1&-1\\0&0\end{array}\right) + y\left( \begin{array}{ccc}0&0\\1&0\end{array}\right) + z\left( \begin{array}{ccc}0&0\\0&1\end{array}\right) = x{\cal M}_1 + y{\cal M}_2 + z{\cal M}_3[/tex]

So [itex]{\cal B}=\left\{\cal M}_1,{\cal M}_2,{\cal M}_{3} \right\}[/itex] forms a basis for [itex]{\cal U}[/itex]. And that it has dimension 3 after showing that they are independent.

Also,

[tex]{\cal W} = \left( \begin{array}{ccc}a&b\\-a&c\end{array}\right)[/tex]

So
[tex]
\left\{\left( \begin{array}{ccc}1&0\\-1&0\end{array}\right),\left( \begin{array}{ccc}0&1\\0&0\end{array}\right),\left( \begin{array}{ccc}0&0\\0&1\end{array}\right)\right\}
[/tex]

is a basis for [itex]{\cal W}[/itex] (with three dimensions).

Any help appreciated. Thanks :smile:
 
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  • #2
Yeah, you just need to show that M1,M2 and M3 are independent.

The wording of question 2 is wrong, but I take it's an analogous question which has an analogous solution.
 
  • #3
Yeah sorry, was slack on that bit. What about the intersection? I get some wacky answer. With a basis; the matrices [tex]\left( \begin{array}{ccc}0&0\\0&1\end{array}\right)[/tex] and [tex]\left( \begin{array}{ccc}1&-1\\-1&0\end{array}\right)[/tex]
 
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What is a vectorspace of matrices?

A vectorspace of matrices is a set of matrices that follow certain rules or properties, such as closure under addition and scalar multiplication. It is a mathematical concept used in linear algebra to study the properties and relationships between matrices.

What are the properties of a vectorspace of matrices?

The properties of a vectorspace of matrices include closure under addition and scalar multiplication, associativity, commutativity, and the existence of an identity and inverse element. Additionally, the set must contain the zero vector and must be closed under matrix multiplication.

How is a vectorspace of matrices different from a regular vectorspace?

A vectorspace of matrices is different from a regular vectorspace in that it consists of matrices instead of individual elements. Additionally, the operations of addition and scalar multiplication are defined for matrices in a different way than for regular vectors.

What are some real-world applications of vectorspace of matrices?

Vectorspace of matrices has many real-world applications, such as in computer graphics, where matrices are used to represent transformations and rotations. It is also used in data analysis and machine learning to represent and manipulate data in a structured way.

How is a vectorspace of matrices used in linear algebra?

Vectorspace of matrices is a fundamental concept in linear algebra, as it allows for the study of linear transformations and systems of linear equations. Matrices can be used to represent linear transformations and solve systems of equations, making them an essential tool in linear algebra.

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