Find a 2x2 Matrix which performs the operation....

[Jtechguy21]

Homework Statement

[/B]Find the matrix that performs the operation

2x2 Matrix which sends e1→e2 and e2→e1

Homework Equations

The Attempt at a Solution

[/B]
I know e1 = < 1 , 0>
and e2 = <0 , 1>

Basically I'm not quite sure what the question is asking. This is the one of the problems I am currently stuck on. He is teaching out of the book-currently

Can someone please explain what this means?

 

[Orodruin]

The problem asks you to find a matrix $A$ such that $A e_1 = e_2$ and $Ae_2 = e_1$.

 

[Jtechguy21]

The problem asks you to find a matrix $A$ such that $A e_1 = e_2$ and $Ae_2 = e_1$.

thanks. So If i understand from your clarification,

This 2x2 matrix that I'm looking for, when I multiply it by e1 it will give me e2, and vice-versa, this unknown matrix multipled by e2 will give me the result e1.

 

[Orodruin]

Yes.

 

[Jtechguy21]

Yes.

Thanks again.

When i find the 2 these two matrixes that perform these two operations. When I multiply them together the result should be the 2x2 matrix that this question is asking for?

 

[Orodruin]

No. You need to find a single matrix which satisfies both relations.

 

[Jtechguy21]

No. You need to find a single matrix which satisfies both relations.

How is that even possible?
1] I found a 2x2 matrix we'll call A that when you multiply it by e1 = e2
2] I also found a 2x2 matrix B that when you multiply it by e2 =e1

But how is it possible to satisfy both conditions at the same time with the same 2x2 matrix? Any hints?

 

[Orodruin]

Any matrix is defined by its action on a complete basis. If you just know how it acts on a single vector you cannot determine it uniquely. I suggest assuming the most general form
$$\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$
and examine what you can say about $a,b,c$ and $d$ from your requirements.

 

[Jtechguy21]

Any matrix is defined by its action on a complete basis. If you just know how it acts on a single vector you cannot determine it uniquely. I suggest assuming the most general form
$$\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$
and examine what you can say about $a,b,c$ and $d$ from your requirements.

Nevermind I just figured it out. thanks for your help with adding to my knowledge/intuition with that statement about defining a matrix actions.
!(took me long enough)

The 2x2 matrix i came up with was the following.

 

[WWGD]

Like Orodruin was saying, a matrix associated to a linear map T is uniquely defined by the effects of T on the given choice of basis. Then , for a map (Assummiing this, given your posts) from $\mathbb R^2$ to itself , both with the same basis, the associated matrix is $[(Te_1)^T , (Te_2)^T]$. where the T means transpose. Note that the copies of $\mathbb R^2$ may have different bases, then the solution is different. There are many variants, of course, of maps between vector spaces ( or even --free, of course-- rings/modules) of different dimensions, with different bases, but the idea of your case generalizes nicely here.