Calculating the dimension of intersection of two matrices

[Mutlu CELIKKOL]

<Moderator's note: Moved from a technical forum and thus no template.>

I am at the beginners level of linear algebra and having problem of the intersection of matrices. Your kind help is much appreciated for the following question

Let\quad M1=\begin{Bmatrix} x & -x \\ y & z \end{Bmatrix},\quad M2=\begin{Bmatrix} a & b \\ -a & c \end{Bmatrix},\quad x,z,y,z\quad a,b,c\quad \in \quad F\\ calculate\quad the\quad following;\quad \\ a)\quad dim\quad M1+M2\\ b)\quad dim\quad M1\quad \cap \quad M2\\

 

[fresh_42]

What do you mean by the intersection of matrices? Do you mean $\mathbb{R}\cdot \begin{bmatrix}x&-x\\y&z\end{bmatrix} \cap \mathbb{R}\cdot \begin{bmatrix}a&b\\-a&c\end{bmatrix}$ or what is it?

 

[Mutlu CELIKKOL]

What do you mean by the intersection of matrices? Do you mean $\mathbb{R}\cdot \begin{bmatrix}x&-x\\y&z\end{bmatrix} \cap \mathbb{R}\cdot \begin{bmatrix}a&b\\-a&c\end{bmatrix}$ or what is it?

That is exactly what I mean

 

[fresh_42]

So you should start to compare them. You have $a_{11}=-a_{12}$ and $a_{11}=-a_{21}$. What do you get from that?

And please post those kind of questions in the future in our homework section, including the use of the (automatically inserted) template!

 

[Mark44]

What do you mean by the intersection of matrices? Do you mean $\mathbb{R}\cdot \begin{bmatrix}x&-x\\y&z\end{bmatrix} \cap \mathbb{R}\cdot \begin{bmatrix}a&b\\-a&c\end{bmatrix}$ or what is it?

What does this notation mean? Particularly $\mathbb{R}\cdot \begin{bmatrix}x&-x\\y&z\end{bmatrix}$?

 

[fresh_42]

It was just a suggestion of a possible interpretation, the straight line through the given matrix as subspace of $\mathbb{M}(2,\mathbb{R})$.

 

[Mark44]

It was just a suggestion of a possible interpretation, the straight line through the given matrix as subspace of $\mathbb{M}(2,\mathbb{R})$.

I'm even more lost now.

What is a "straight line through the given matrix"?
What is $\mathbb{M}(2,\mathbb{R})$?
I'm familiar with notations such as $\mathbb{M}_{2, 3}$ or the like, for matrices with 2 rows and 3 columns, or $\mathbb{M}_{m, n}$, for m x n matrices. In both examples, the field is unstated.

 

[fresh_42]

I like to note the field as it is often important here whether the reals, complex numbers or even a finite field is allowed. With only one index, the quadratic version is meant, so $\mathbb{M}(2,\mathbb{R})$ means all real $2\times 2$ matrices. They build a vector space and one matrix, as given by the OP is a vector therein. Thus there is also a line through this vector and the zero matrix, which defines a one dimensional subspace. And the intersection is a point in this space.

Another possibility would have been $\begin{bmatrix}x&-x\\y&z\end{bmatrix} = \left\{ \begin{bmatrix}a&b \\ c&d\end{bmatrix} \in \mathbb{M}(2,\mathbb{R}) \, : \, b=-a \right\}$ in which case we get a three dimensional subspace.

Both are possible and the wording in post #1 doesn't tell. I used the line interpretation as it was easy to type. My real goal was to provoke a clarification.