Question

Suppose L: R^3 maps R^3 is a matrix transformation. and L((0,2,-1))=(-1,1,-4) and L((0,1,-1))=(1,-5,5).

Find
L(3(0,2,-1)-2(0,1,-1) =
L((0,-5,3)) =

Help please!

Answer 1

L is linear, so what does that tell us about scalars?

Answer 2

There scalable?

Answer 3

L(kx) for some constant k is equal to: kL(x)

Answer 4

Right, that's the formula for matrix transformations. Not sure how to proceed with it to solve the problem

Answer 5

also, a linear property is such that L(x+y) = L(x)+L(y), putting this to your problem we get:

L(3(0,2,-1)-2(0,1,-1)

L(3(0,2,-1)) + L(-2(0,1,-1) )

3 L(0,2,-1) -2 L(0,1,-1)

Answer 6

and they define the L parts for those ...

Answer 7

now, a trickier part may be to find 0,-5,3 as a linear combination of the vectors being Ld

Answer 8

How did you get from the 2nd one to the last one?

Answer 9

pulled out the scalars of course

Answer 10

Oh yeah i see it now, but that's all there is to the solution? I thought it would be a lot harder

Answer 11

you have presented to you:

L(x) = u, L(y) = v, and for some sclaras, j,k

L(kx + jy) = ??

L(kx) + L(jy)

kL(x) + jL(y)

ku + jv

Answer 12

now the question becomes trickier for the 2nd one:

find kx + jy = (0,-5,3) and then apply the operations from the first part

Answer 13

Awesome it became a lot easier to understand, thanks!

Answer 14

youre welcome :)