How Do You Compute the Matrix Representation and Check for Eigenbasis?

[Punkyc7]

For each of the following linear operators T on a vector space V and ordered bases beta, compute [T]beta, and determine whether beta is a basis conisting of eigen vectors of T.

V=R^2, T((a,b)^t)= (10a-6b
17a-10b)

and beta ={(1,2)^t , (2,3)^t)


im using transpose because I am not sure how to make thenbe vectors going down


My question is how do you do it. My back doesn't give any example on how to these types of problems. I thought of using e1 and e2 but that doesn't get me the answer in the back of the book and i tried plugging in the basis but that didnt work.


If it helps the answer for [T]beta =(02
-10)
but i have no idea how they got that

 

[micromass]

Hi PunkyC7!

What is [T]beta? Is it the matrix of T with respect to beta?

If so, then denote beta={x,y}. You will have to calculate T(x) and T(y) and write these things as a linear combination of x and y. Then you need to put it into a matrix.

If it isn't clear, I'll give an example.

 

[Punkyc7]

yeah its with respect to beta

so if you plug in our first beta we get (-2,-3)^t do we set that equal to x(1,2)+y(2,3)?

 

[micromass]

yeah its with respect to beta

so if you plug in our first beta we get (-2,-3)^t do we set that equal to x(1,2)+y(2,3)?

Well, you want to find x and y such that (2,3)=(x+2y,2x+3y). This is a system of two equations and two unknowns...

 

[Punkyc7]

so that's how they go (0,-1) ok
so know do you know if it consists of eigen vectors

 

[micromass]

Well, what is an eigenvector?

 

[Punkyc7]

an eigen vector is a vector is a vector in V that is non zero such that T(v)=lamdav where lamda is the eigen value

 

[micromass]

Indeed, so is T(1,2) from the form lambda*(1,2)?

 

[Punkyc7]

no i guess not so, since we don't have any other vector to make it span R^2 there is know reason to check the other right

 

[micromass]

Certain? What is T(1,2) in terms of (1,2) and (2,3)?

 

[Punkyc7]

x=0
y=-1

 

[micromass]

So T(1,2)=-(2,3), so this indeed means that you won't have a basis of eigenvectors!

 

[Punkyc7]

why does it mean it?

 

[micromass]

Because T(1,2) is not a multiple from (1,2). That is T(1,2)=-(2,3), and thus not T(1,2)=lambda*(1,2)...

 

[Ray Vickson]

You can express e1 = <1,0> and e2 = <0,1> in terms of beta[1] and beta[2]. Then anything of the form a*e1 + b*e2 can be re-written as a linear combination of beta[1] and beta[2].

RGV