Linear transformation, basis of the image

[Chris1557]

Homework Statement

From Calculus we know that, for any polynomial function f : R -> R of degree <= n, the function
I(f) : R -> R, s -> ∫₀^s f(u) du, is a polynomial function of degree <= n + 1.

Show that the map
I : Pn -> Pn+1; f -> I(f),
is an injective linear transformation, determine a basis of the image of I and find the matrix
M in M_(n+2)x(n+1)(R) that represents I with respect to the basis 1,t,...,tn of P_n and the basis 1,t,...,tⁿ⁺¹ of P_n+1.

Homework Equations

The Attempt at a Solution

I found a topic involving setting f(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + ... + a₁x + a₀, and g(x) = b_nxⁿ + b_n-1x^n-1 + b_n-2x^n-2 + ... + b₁x + b₀.
Then show that L(f + g) = L(f) + L(g) and that aL(f) = L(ax).

That seems ok, but I have no idea how to determine a basis of the image and I am confused on what the final part in notating and what to do.

Thanks.

 

[micromass]

I found a topic involving setting f(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + ... + a₁x + a₀, and g(x) = b_nxⁿ + b_n-1x^n-1 + b_n-2x^n-2 + ... + b₁x + b₀.
Then show that L(f + g) = L(f) + L(g) and that aL(f) = L(ax).

That's a very good start. This will show that we are dealing with a linear transformation.

That seems ok, but I have no idea how to determine a basis of the image and I am confused on what the final part in notating and what to do.

Let's first determine injectivity. For this, we take a basis of P_n (can you find an easy basis for this?). Take the image of this basis under I and determine whether this image is linear dependent. If so, the function is injective, and behold: the image of the basis is (in that case) a basis for the image!

 

[Chris1557]

Let's first determine injectivity. For this, we take a basis of P_n (can you find an easy basis for this?). Take the image of this basis under I and determine whether this image is linear dependent. If so, the function is injective, and behold: the image of the basis is (in that case) a basis for the image!

For the easy basis of P_n can we just use xⁿ, x^n-1,..., x, 1

How do we take the image of the basis under I?

The matrix part is also confusing me.

 

[micromass]

Just calculate I(xⁿ),...,I(x),I(1). This should be an easy calculation...