Image of Linear Transformation with Given Vectors

In summary, the linear transformation T:R2R2 can be represented by T(x y)= [-9x-9y, 9x+y] and follows the rule T(u+v)= T(u)+ T(v). To find the image of an arbitrary vector (x y), you can use the formula T(u)= AT(v1)+ BT(v2) where A=-x-y and B=-2x-y. Then, the image would be T(x y)= [-9x-9y, 9x+y].
  • #1
doublemint
141
0

Homework Statement



Let v1=
1
-2
and v2=
-1
1


Let T:R2R2 be the linear transformation satisfying
T(v1)=
9
7
and T(v2)=
0
-8


Find the image of an arbitrary vector
x
y



Homework Equations





The Attempt at a Solution



I thought it might have to do something with T(u+v)=T(u)+T(v) or some sort of transformation, but I cannot seem to get it...
Any help would be appreciated!
Thanks!
 
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  • #2
You are given that
[tex]T\left(\begin{bmatrix}1 \\ -2\end{bmatrix}\right)= \begin{bmatrix} 9 \\ 7\end{bmatrix}[/tex]
and that
[tex]T\left(\begin{bmatrix}-1 \\ 1\end{bmatrix}\right)= \begin{bmatrix}0 \\ 8\end{bmatrix}[/tex]

And you want to determine
[tex]T\left(\begin{bmatrix} x \\ y\end{bmatrix}\right[/tex]

Yes, you want to use T(u+v)= T(u)+ T(v). Specifically if [itex]u= Av_1+ Bv_2[/itex] then T(u)= AT(v_1)+ BT(v_2). So first you want find A and B such that
[tex]\begin{bmatrix}x \\ y \end{bmatrix}= A\begin{bmatrix}1 \\-2\end{bmatrix}+ B\begin{bmatrix}-1 \\ 1 \end{bmatrix}[/tex]
 
  • #3
Alright, so I got
A=-x-y
B=-2x-y
I'm guessing then we follow through with T(u)= AT(v_1)+ BT(v_2),

T(x y)=[T(1 -2)T(0 -8)][A B]=[9A, 7A-8B]

Then I sub in A and B:

[9(-x-y), 7(-x-y)-8(-2x-y)]= [-9x-9y, 9x+y]

Is this what I was supposed to do? I think now I have to factor out the x-y, but I can't do it to 9x+y. Did I do something wrong at finding A and B?
 
  • #4
I just submitted my work, it was right after all!
Thanks HallsofIvy!
 

What is a linear transformation image?

A linear transformation image is a mathematical concept that describes how a set of points in one space gets mapped to another space. It is a function that takes in a vector and outputs a new vector, following certain rules and properties.

What are the properties of a linear transformation image?

The properties of a linear transformation image include preservation of addition (T(u+v)=T(u)+T(v)), preservation of scalar multiplication (T(cu)=cT(u)), and preservation of the zero vector (T(0)=0). It also preserves parallel lines and the origin.

What is the difference between a linear transformation and a non-linear transformation?

A linear transformation follows the properties mentioned above, while a non-linear transformation does not. This means that a non-linear transformation does not preserve parallel lines or the origin, and does not follow the same rules for addition and scalar multiplication.

How is a linear transformation image represented?

A linear transformation image is typically represented by a matrix. The columns of the matrix represent the image of the standard basis vectors in the original space, and the rows represent the coordinates of the transformed vectors in the new space.

What are some real-world applications of linear transformation images?

Linear transformation images have many applications in fields such as computer graphics, engineering, and physics. They are used to represent and manipulate 3D objects, perform image transformations in digital image processing, and model physical systems in engineering and physics simulations.

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