linear transformation question. i cant figure it out?

Question

v1=[-3]
[-4]

v2=[-2]
[-3]

Let T; R^2 -> R^2 be the linear transformation satisfying

T(v1)= [12]
[-32]
and
T[v2]=[7]
[-22]

Find the image of an arbitrary vector
[x]
[y]

T[x]=[ _ _ ]
[y]=[ _ _ ]

Answer 1

Muhahahaha. This is *exactly* the kind of problem I used to give my students when I taught Linear Algebra ☺

OK. You know that T[v1] = [12,32] (and I'll write it as a row vector instead of a column because it's easier ☺)
What do you *immediately* know? If you assume T to be of the form
|a...b|
|c...d|
then
a*(-3) + b*(-4) = 12
c*(-3) + d*(-4) = -32
And, from T[v2] = [7,-22] you get
a*(-2) + b*(-3) = 7
c*(-2) + d*(-3) = -22
Now, take the equations in a and b to get

a*(-3) + b*(-4) = 12
a*(-2) + b*(-3) = 7
which will let you solve for a and b. Likewise
c*(-3) + d*(-4) = -32
c*(-2) + d*(-3) = -22
will let you solve for c and d.

Now I'll let *you* do the arithmetic to get a,b,c, and d to find the transform
|a...b|
|c...d|
and multiply it by [x,y] to get your final answer.
That'll also hellp you understand what's going on (and it will develop character ☺)

And tell your math teacher I like him or her already ☺


Doug

Answer 2

ok, right here we bypass. A) enable's say you have a factor (2,3). X=2, Y=3. to alter that to (-y,x), you need to placed -3 interior the 1st slot, then 2 interior the Y slot. (-3, 2). Graph-smart, you're turning the F shape ninety ranges counterclockwise (that's going to look like an F on its area with the little prongs pointing up; the flat area would be alongside the x-axis) B) you're making a vertical stretch. Multiply the Y in each and each factor (x,y) via 2. as an occasion, (2,3) would grow to be (2,6). Graphically, the F will bypass greater up and farther down on the Y axis yet stay the comparable width. think of tall and skinny. C) right here, (2,3) would grow to be (2+3, 3) or (5, 3). Graph-smart, the F will slant. The factors you need to finally end up with are (one million,one million) (2,one million) (0,0) (one million,0) and (-one million,-one million). D) right here you're purely reflecting around the X axis, turning the F upside-down. Make all the Y's on your factors adverse - (2,3) would grow to be (2,-3) E) right here you replicate around the line y=x. certainly, make all the numbers on your factors adverse. (2,3) turns into (-2,-3). Your F will look like that's been circled a hundred and eighty ranges, or upside-down and backwards. desire this enables and stable success!