Question

hello
can anyone help me solve this?
the questions in english are :
a) show that F is linear
b) determine the associated matrix relative to the basis A and B . Is this matrix invertible? yes, give the reverse
c) determine by hand:
the image of the vector
the polynomial Px
d)determine the kernel ker(f) and image Im(f)

Answer 1

To show F is linear, we need show F(aP + Q)(x) = aF(P(x) +F(Q(x)
YOu can break it out by doing 2 parts : F(P + Q)(x) = F(P)+F(Q) and F(aP(x))= aF(P(x))
I do the first one
Let P, Q in R^2[x], a in R
then aP (x) + Q(x) = (aP+Q)(x)
Now, apply F on
\(F(aP + Q)(x) = ((aP+Q)(0), (aP+Q)'(1),\int_0^1 (aP+Q)(x))\)

Answer 2

All you need is "translate" the right hand side to \(F(aP(x)) +F(Q(x)\)

Answer 3

for the first term: \((aP + Q)(x)\) , the property of function give us \((aP+Q)(x) = aP(x) +Q(x)\)
It likes (f+g)(x) = f(x) +g(x)

Answer 4

the second term is derivative, and we know that derivative of the sum = sum of derivative, right? like (f+g)' = f'+g'
hence we have \((aP+Q)'(1) = (aP)'(1) +Q'(1)\)

Answer 5

the last term is integral, exactly the same, we still have
\[\int_0^1 (aP+Q)(x)dx=\int_0^1aP(x) dx +\int_0^1 Q(x) dx\]

Answer 6

Now, combine all

Answer 7

You know that if I have vectors
\[\left[\begin{matrix}a +b \\c +d\end{matrix}\right] =\left[\begin{matrix}a \\c \end{matrix}\right]+\left[\begin{matrix}b \\c \end{matrix}\right]\], right?

Answer 8

hence, your result can be broken down like that

Answer 9

\[\left[\begin{matrix}aP(x) + Q(x)\\aP'(1) +Q'(1)\\\int_0^1aP(x)dx +\int_0^1Q(x)\end{matrix}\right]=\left[\begin{matrix}aP(x) \\aP'(1) \\\int_0^1aP(x)dx \end{matrix}\right]+\left[\begin{matrix} Q(x)\\Q'(1)\\\int_0^1Q(x)\end{matrix}\right]\]

Answer 10

From the right, the first matrix is F(aP(x), and the second one is F(Q(x)
Done, right?

Answer 11

ok that seems quite logical , i really got confused by the writing

Answer 12

Of course, you must jot down in neat. I inserted the explanation when solving it. It makes the proof has many irrelevant terms. hehehe...

Answer 13

okay thank you , and do you know b, c and d? ( I'm very grateful that you help me)

Answer 14

For b) matrix A is invertible since its determinant =-12 \(\neq 0\)

Answer 15

So does B, det(B) = 2

Answer 16

oh but the question is to combine those two bases to get the associated matrix

Answer 17

And you know how to find their inverse, right?

Answer 18

yes the inverse and determinant aren't the problem , that's just math but i need the matrix that is formed by those two bases given F

Answer 19

I don't get the question.

Answer 20

it's called the associated matrix in dutch. But i can't find a decent translation for it

Answer 21

You meant the transformation that changes base A to base B??

Answer 22

yes i think so

Answer 23

so, just put [B|A] and try to change to [Identity| required matrix] by rref. Dat sit

Answer 24

this is an example of what we did on another exercise