How to Express a Vector as a Linear Combination of Eigenvectors?

[Angello90]

Hey guys,

I'm studing to my exams now, and I came accors this question i eigenvectors where you find them and bla bla. There is part to it which asks to express vetor

X= [2/1]

as a linear combination of eigenvectors. Hence calculate B2X, B3X, B4X and B51X, simplifying your answers as much as possible.

How do you do the linear combination?

Thanks!

 

[HallsofIvy]

The "bla bla" does not help at all. Perhaps you could post the entire question?

For example, what does "[2/1]" mean? Is that a two dimensional vector with components 2 and 1? Write it as a linear combination of what eigenvectors? Is there some matrix or linear transformation you haven't mentioned? And what are " B2X, B3X, B4X and B51X"? Those are not standardized notations.

 

[Angello90]

Ok sorry I suppose I didn't make myself clear.

I have two eigenvectors.

First: Look at 1.jpg
Second : Look at 2.jpg

The vector X (look at 3.jpg) is to be written as a linear combination of eigenvectors. How do you do that? It's just a theory I'm interested in not solution to the question.

Thanks and sorry for inconvenience.

 

[jrlaguna]

OK, now your question is clear. Let v1=(-3,1) be the first eigenvector, and v2=(-2,1) the second one. Now you want "a1" and "a2" such that

X = a1 v1 + a2 b2

That is equivalent to solving a linear system

[2] = [-3 -2] [a1]
[1] [ 1 1] [a2]

Where the eigenvectors went as columns.

 

[Angello90]

So the answer should look like this (look at the ans.jpg)?
And are a1 and a2 variables?

Thanks a lot!

 

[jrlaguna]

correct! :)

 

[Angello90]

Thanks a lot jrlaguna!