Re: Linear Algebra: Linear Independence of Eigenvectors

Wanshan Li wrote:
Let V1 and V2 be eigenvectors of a linear operator T on R^n, and let t1 and t2, respectively, be the corresponding eigenvalues. Prove that if t1 is not equal to t2, then {V1, V2} is linearly independent.

Thanks.


Wanshan Li

What does it mean for two vectors to be linearly independent?
If a pair of vectors fails to be linearly independent, is
there any conclusion you can draw?

How might this be related to the fact that you're dealing
with eigenvectors?

There are two directions you might consider:

First, in the direction the problem was posed:

Given

T V1 = t1 V1
T V2 = t2 V2 with t1 =/= t2

and

a1 V1 + a2 V2 = 0

show

a1 = 0 & a2 = 0.

Alternatively, you might prove this:

Let

T V1 = t1 V1
T V2 = t2 V2

and suppose V1 & V2 are NOT linearly
independent. Show t1 = t2.

Dale
.

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