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!
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!
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 R^{2}, 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?