Question

question:

Answer 1

Let L: \[R ^{2} -> R ^{2} \] be the linear transformation defined by
\[L[\left(\begin{matrix}a \\ b\end{matrix}\right)]= \left[\begin{matrix}1 & 2 \\ 2 & 4\end{matrix}\right] \left(\begin{matrix}a \\ b\end{matrix}\right)\]
a)find ker L
b) find a set of vectors spanning range L.

Answer 2

just explain the procedure how to do it. @lalaly

Answer 3

what does ker mean

Answer 4

i guess its just L . ker is a printing mistake sorry for that uncle

Answer 5

?

Answer 6

kernel

Answer 7

maybe i m sure about kernel its only written ker L

Answer 8

i just need the way to solve it

Answer 9

I'm looking over my notes, I always forget this stuff

Answer 10

no problem

Answer 11

oh, kernel is the same thing as null space for a linear transformation
i.e. the set of all \(\vec x\) in \(A\vec x=\vec 0\) is the kernel, so just solve that matrix to find part a)

Answer 12

ok

Answer 13

that is, solve\[ \left[\begin{matrix}1 & 2 \\ 2 & 4\end{matrix}\right] \left(\begin{matrix}a \\ b\end{matrix}\right)=0\]

Answer 14

then we'll get two eqs right ???

Answer 15

yes, from which we will get the two vectors that will make up the basis of our kernel

Answer 16

at least I think that is what they want

Answer 17

ok

Answer 18

about what about part B ???

Answer 19

consulting notes again...

Answer 20

ok

Answer 21

hm... I'm perhaps a bit confused because my notes don't seem to agree with wikipedia.
I think the dimension of the null space, i.e. the nullity is the kernal
in that case, the answer to part a) would be the number of vectors that make up the basis for the null space of your transformation matrix

the answer to part b) must then be the basis itself I guess, which is the set of vectors that span \(A\vec x=\vec0\)

Answer 22

ok thank you i got it :) i just need some practice now

Answer 23

yeah, me too :)
welcome, good luck