Question

if the eigenspace is reduced to the matrix:
|10|
|01|
what would the basis be or v1? wouldn't it just be v1=[0,0]

Answer 1

yep pretty much since

\[\left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]\left(\begin{matrix}x _{1} \\ x _{2}\end{matrix}\right)=\left(\begin{matrix}0 \\ 0\end{matrix}\right)\]

Answer 2

so here x1+0x2=0 and 0x1+x2=0

we typically just use one equation to determine its eigenvector

Answer 3

yeah that what i thought, but i got it wrong in my quiz

Answer 4

but thanks for clearing it up

Answer 5

so its wrong?

Answer 6

the question was: Define T: R2 right arrow R2 by T(x) = Ax, where
A =

1 −6

2 −6
. Find a basis B = {v1, v2 } for R2 with the property that [T]B is diagonal.

Answer 7

i found the eigen vectors to be -1 and -5

Answer 8

and when i put them into the matrix to get the eigen vectors, i got the reduced matrix as above,
|10|
|01|

Answer 9

sorry i mean eigen values to be -1 and -5

Answer 10

i don't think they are your eigen values..

Answer 11

you should have eigen values -3 and -2

Answer 12

do you want to go through this?

Answer 13

oh right, no i just re did it

Answer 14

yeah made a mistake with the characteristic equation, thanks for clearing it up!

Answer 15

yep sweet. just make sure you carefully solve for the determinant. critical step where everyone can get it wrong and hence your solutions will be wrong

Answer 16

you can do it here if you want so i can see your thought process