Linear transformations

• Oct 12th 2008, 08:09 PM
Apprentice123
Linear transformations
1) Verify that the following transformations are linear

a) $T:R^2=>M(2,2)$
$T(x,y)=\begin{pmatrix} 2y & 3x \\ -y & x+2y \end{pmatrix}$
How can I solve?

b) $T:R^2$
$T(x,y)=(x^2+y^2,x)$
Not is Linear transformations:
$f(u+v)=((x1)^2+2x1x2+(x2)^2+(y1)^2+2y1y2+(y2)^2,x1 +x2)$

c) $T:R^3=>R^3$
$T(x,y,z)=(x+y,x-y,0)$
Yes this is Linear transformation

d) $T:R=>R^2$
$T(x)=(x,2)$
Yes this is Linear transformation

How to handle first? The rest this correct?
• Oct 12th 2008, 09:06 PM
ThePerfectHacker
Quote:

Originally Posted by Apprentice123
1) Verify that the following transformations are linear

a) $T:R^2=>M(2,2)$
$T(x,y)=\begin{pmatrix} 2y & 3x \\ -y & x+2y \end{pmatrix}$
How can I solve?

Remember what linear means.
You need to show $T(\bold{x}+\bold{y}) = T(\bold{x}) + T(\bold{y})$.

Now if $\bold{x} = (x_1,x_2)$ and $\bold{y} = (y_1,y_2)$ does $T((x_1+y_1,x_2+y_2)) = T(x_1,x_2)+ T(y_1,y_2)$.

And you also need to show $T(k\bold{x}) = kT(\bold{x})$ in a similar way.
• Oct 13th 2008, 06:33 AM
Apprentice123
Quote:

Originally Posted by ThePerfectHacker
Remember what linear means.
You need to show $T(\bold{x}+\bold{y}) = T(\bold{x}) + T(\bold{y})$.

Now if $\bold{x} = (x_1,x_2)$ and $\bold{y} = (y_1,y_2)$ does $T((x_1+y_1,x_2+y_2)) = T(x_1,x_2)+ T(y_1,y_2)$.

And you also need to show $T(k\bold{x}) = kT(\bold{x})$ in a similar way.

I know this was what I did. And about my doubts?
• Oct 13th 2008, 07:42 AM
ThePerfectHacker
Quote:

Originally Posted by Apprentice123
I know this was what I did. And about my doubts?

$T(x_1,x_2) = \begin{bmatrix}2x_2&3x_1\\-x_2&x_1+2x_2\end{bmatrix}$

$T(y_1,y_2) = \begin{bmatrix}2y_2&3y_1\\-y_2&y_1+2y_2\end{bmatrix}$

$T(x_1,x_2)+T(y_1,y_2) = \begin{bmatrix} 2(x_2+y_2) & 3(x_1+y_1) \\ - (x_2+y_2) & (x_1+y_1) + 2 (x_2+y_2) \end{bmatrix} = T(x_1+y_1,x_2+y_2)$.

Thus, the first condition is true.
Proving the second one is a lot easier.