Question

Let T be a function from R2 to R2 such that T(2,1)=(1,0) and T(1,-1)=(2,1). Find T(3,-4)

Answer 1

I don't think you have enough information to predict where that would go. Any point in R2 would do, if you have no more specifications than you've given here.

Answer 2

would the fact that T is a linear function help? because besides that, this is all that is given

Answer 3

a(2,1)+b(2,-1)=(3,-4)

We need to find constants a and b
2a+2b=3 => 2a+2b=3
1a-1b=-4 => -2a+2b=8
---------------
0+4b=11
=> b=11/4
So what is a?

Answer 4

a=-5/4, but how did you come up with the equation a(2,1)+b(2,-1)

Answer 5

oops that second pai rwas suppose to be (1,-1) my bad

Answer 6

\[a(2,1)+b(1,-1)=(3,-4)\]

Answer 7

my eyes were playing tricks on me

Answer 8

ok i see now

Answer 9

So we need to solve this for a and b

Answer 10

one sec i'm figuring this as we go to
i know that is the first step

Answer 11

oh wait i think i'm fixing to work out the other details

Answer 12

\[(3,-4)=a(2,1)+b(1,-1)\]
2a+b=3
a-b=-4
i get a=7/3 and b=19/3

Answer 13

@zarkon I forgot what to do here after this part

Answer 14

i know it is something stupid i'm forgetting

Answer 15

not a prob, i actually dnt know what to do either

Answer 16

T is linear

Answer 17

I assume

Answer 18

yup

Answer 19

then T(av+bw)=aT(v)+bT(w)

Answer 20

lost my connection to the site for a sec

Answer 21

wait i got it i think don't say it zarkon please

Answer 22

will you check me though lol

Answer 23

\[T(3,-4)=T(a(2,1)+b(1,-1))=T(a(2,1))+T(b(1,-1))=aT(2,1)+bT(1,-1)\]
\[=a(1,0)+b(2,1)\]
Replace a with what you found
Replace b with what you found
The answer will be this with your replacements
=(a+2b,b)

Answer 24

Right Zarkon?
:)

Answer 25

yes

Answer 26

:) I love you zarkon your the best
platonic love of course

Answer 27

that's what i got too, but the teacher marked it wrong was confused. and then T(2,1)=(1,0). with the (a+2b, b), if you let a=2 and b=1. T(2,1)=(4,1) which does not equal with what was given? why is it?

Answer 28

so you used the a and b you found above?

Answer 29

I didn't check if your arithmetic was right. I could.

Answer 30

yeah i think i see something wrong

Answer 31

2a+b=3
a-b=-4

-------
3a=-1
a=-1/3

Answer 32

-1/3-b=-4
-b=-4+1/3
-b=-12/3+1/3
-b=-11/3
b=11/3

Answer 33

\[T(x,y)=\left(x-y,\frac{x-2y}{3}\right)\]

if you want to find the image of any vector \(v=(x,y)\)

Answer 34

here is what i did. given T(2,1)=(1,0) and T(1,-1)=(2,1)
i let there be a X=(a,b) which is in R2
X=a(2,1)+b(1,-1)
TX=aT(2,1)+bT(1,-1)=a(1,0)+b(2,1)
so T(a,b)= (a+2b,b)
T(3,-4)=(-5,-4)
sorry I took so long,

Answer 35

T(3,-4)=(7,11/3)

Answer 36

use the a,b myininaya found

Answer 37

i must have done some error in the arithmetic one the way

Answer 38

or use my general formula ;)