Is the zero Matrix a vector space?

In summary, the conversation discusses finding a base for vector spaces M, N, M + N, and M ∩ N, where M and N are matrices with elements from the real numbers. The attempted solution proposes a base for the first three vector spaces, but there are discrepancies in the proposed base and the chosen scalar field is not specified for the fourth vector space. Further clarification is needed to determine if the proposed base works for M ∩ N.
  • #1
Alex Langevub
4
0

Homework Statement


So I have these two Matrices:
M = \begin{pmatrix}
a & -a-b \\
0 & a \\
\end{pmatrix}
and
N =
\begin{pmatrix}
c & 0 \\
d & -c \\
\end{pmatrix}

Where a,b,c,d ∈ ℝ

Find a base for M, N, M +N and M ∩ N.

Homework Equations


I know the 8 axioms about the vector spaces.

The Attempt at a Solution


I chose these fours matrices as a base for the first three vector spaces.
\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}

I got that the only vector space satisfying M ∩ N is
\begin{pmatrix}
0 & 0 \\
0 & 0 \\
\end{pmatrix}

If M ∩ N constitutes a vector space, I can use the same base as the other three. But if it doesnt, I have to explain why.

Not sure how to go about that...
 
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  • #2
Does ##V=\{\vec{0}\}## fulfill the axioms of a vector space?

I don't understand what you wrote. ##M(a,b)=\begin{bmatrix}a & -a-b \\ 0& a\end{bmatrix}## is probably meant to be ##V_M=\{M=M(a,b)\,\vert \,a,b \in \mathbb{R}\}##. So how can ##\begin{bmatrix}1&0\\0&0\end{bmatrix}## be in ##V_M## or in ##V_N\,?## In ##V_M## there are only matrices with a diagonal with two identical entries.

Also the choice of the scalar field plays a role in the third case ##V_{M\cap N}##. You should mention it, because there are two different solutions, depending on the kind of field. I assumed you meant the reals, but this isn't clear.
 

1. What is the definition of a vector space?

A vector space is a mathematical structure that consists of a set of vectors, along with operations such as addition and scalar multiplication, that satisfy a set of axioms. These axioms include closure under addition, scalar multiplication, and distributivity, among others.

2. Can the zero matrix be considered a vector space?

Yes, the zero matrix can be considered a vector space. It satisfies all of the axioms of a vector space, including closure under addition, scalar multiplication, and distributivity. It also contains the zero vector, which is a requirement for a vector space.

3. How is a zero matrix defined?

A zero matrix is a matrix where all of the elements are equal to zero. It is often denoted by the symbol O or 0.

4. What are some examples of vector spaces?

Some examples of vector spaces include the set of real numbers, the set of complex numbers, and the set of polynomials with real coefficients. Additionally, the set of all n-dimensional vectors is also a vector space.

5. How can the zero matrix be useful in linear algebra?

The zero matrix can be useful in linear algebra as it serves as the additive identity element, meaning that when added to any other matrix, it does not change the result. It also helps to define concepts such as linear independence and span in vector spaces.

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