Question

Let T: R3 →R3 be a Linear transformation such that
T (2e1 – e2 = 5e3 ) =[3 -1]
T( -e1+e2+5e3) = [ 2 -3]
T( e1 – e2 + e3) = [ 2 1]
Find the matrix A of transformation such that T(x) = A x

Answer 1

The columns of A are T(e1), T(e2), T(e3).

Answer 2

tx

Answer 3

ok, I don't get it how you find A inverse?

Answer 4

the first matrix?

Answer 5

I think you made a mistake loser66. If we add the 2nd and the third equation we have
\[
T(6e3)=[4, -2]\Leftrightarrow T(e3)=[2/3 ,-1/3]
\]

Answer 6

Go ahead :)

Answer 7

tx, yes I got the first part

Answer 8

right

Answer 9

right

Answer 10

ok

Answer 11

- T (e1) + T (e2) + 5 Te(3) = [ 2 -3]

Answer 12

T e(1) - T (e2) + T(e3) = [2 1]

Answer 13

I'll be right back,,,thank you

Answer 14

Be careful loser66. The map is from R^3 to R^2. Hence A will be a 2 by 3 matrix

Answer 15

You should write [3 -1], [2 -3] and [2 1] on your matrix as columns

Answer 16

We don't have to find the invers to solve this problem. The idea is to write e1,e2,e3 as linear combination of 2e1 – e2 = 5e3, -e1+e2+5e3 and e1 – e2 + e3

Answer 17

yes, you are right that should be a plus

Answer 18

I misscopied it

Answer 19

Before you leave, your mistake is to treat T(e1), (e2), T(e3) as number and you put them as one column vector

Answer 20

I'm here

Answer 21

yes, they are vector so in this case we can't put it as
\[
\begin{pmatrix}T(e_1)\\T(e_2)\\T(e_3)\end{pmatrix}
\]

Answer 22

you guys are great

Answer 23

To find \(T(e_1)\) first we need to find \(a,b,c\in R\) such that
\[
e_1=a(2e1 – e2 + 5e3)+b( -e1+e2+5e3)+c( e1 – e2 + e3)
\]
once we know \(a,b,c\) then by linearity we can compute T(e1). The same goes gor T(e2) and T(e3)

Answer 24

I am afraid I am not as generous as you are :). I think that is enough hint to do the problem.

Answer 25

alittle bit more plz

Answer 26

if we simplify the right hand side by collecting similar term we have
\[
e1=(2a-b+c)e1+(-a-b-c)e2+(5a+5b+c)e3
\]
Now comparing the left and the right we have 2a-b+c=1, -a-b-c=0 and 5a+5b+c=0. So you have a system of equation and you can solve it using gaussian elimination.