# Thread: How to show this is Linear? (Urgent!)

1. ## How to show this is Linear? (Urgent!)

I have this Linear Transformation $\displaystyle f: P_3(R) \rightarrow R^2$ given by

$\displaystyle f(P(x))=(P(1),P(-2))$

Then I have to show that $\displaystyle f$ is linear.

I do not really have any idea... SO really need some help please!

2. Show that the following is true.
$\displaystyle f\left( {\alpha P(x) + Q(x)} \right) = \alpha f\left( {P(x)} \right) + f\left( {Q(x)} \right)$

3. Originally Posted by Plato
Show that the following is true.
$\displaystyle f\left( {\alpha P(x) + Q(x)} \right) = \alpha f\left( {P(x)} \right) + f\left( {Q(x)} \right)$
I can show that the right side is: $\displaystyle k= \alpha$

$\displaystyle k(P(1), P(-2))+(Q(1), Q(-2))$

But I do not really know what do with the left side...

4. Originally Posted by lo2
I can show that the right side is: $\displaystyle k= \alpha$

$\displaystyle k(P(1), P(-2))+(Q(1), Q(-2))$

But I do not really know what do with the left side...
That makes no sense at all! f is supposed to map into $\displaystyle R^2$ but you have 3 components! In $\displaystyle R^2$ addition is defined by (a,b)+ (c,d)= (a+c, b+d), "component wise".

f is a function from $\displaystyle P_3(R)$ which, I think, is the space of polynomials of degree 3 or less with real coefficients, to R2, the space of pairs of real numbers. f(P(x)) is defined as the pair (P(1), P(-2)).

So, for example, if [tex]P(x)= x^3+ 2x- 1[/itex], P(1)= 1+ 2- 1= 2 and P(-2)= -8- 4-1= -13. f(P)= (2, -13).

$\displaystyle f(\alpha P+ Q)= (\alpha P(1)+ Q(1), \alpha P(2)+ Q(2))$ and $\displaystyle \alpha f(P)+ Q(P)= \alpha (P(1), P(2))+ (Q(1), Q(2)$. Are those the same?