# Math Help - Linear Algebra Question

1. ## Linear Algebra Question

I'm completly unsure how to tackle this problem

the subject of the following is Linear Transformations.

Given in the text:

....We also have linear transformations on function spaces, such as multiplication operators.
2.5
$
M_f :C^k (I) \to C^k (I),M_f g(x) = f(x)g(x)
$

given that:
$f \in C^k (I),I = [a,b]$ and the operation of differentiation

2.6
$
D:C^{k + 1} (I) \to C^k (I),Df(x) = f'(x)
$

we also have integration

2.7
$
i:C^k (I) \to C^{k + 1} (I),if(x) = \int\limits_a^x {f(y)dy}
$

note also that
2.8
$
D:P_{k + 1} \to P_k
$

and

$
i:P_k \to P_{k + 1}
$

where $P_k$ denotes the space of polynomials in x of degree $\leqslant k$

QUESTION:

compute
$Di$ and $iD$

and specify the null space and range of both i and D

2. It is completely understandable that you may not know that the greatest problem in mathematics communication is the lack of a standard in notation. I dare say that only the authors of your text and their fellows can read that notation. So if you want help, you must explain what those terms and symbols mean.

3. I have edited my question as best as possible. I am still lost on this question. any advice would be of help.

4. The polynomial space of degree k has k+1 basis vectors.

to construct a matrix transformation we Transform each of the basis.

See the below attachments.

I did an example D the derivative transformation on $p_2$

note D goes from P k+1 to P k and

i goes from P k to p k+1

5. ## Cont.

Now you would need to do the same thing for

i the integral transform,

Then compute the product of the two matricies.

I hope this helps.

6. is the null space of the D transformation you calculated any point on the z axis? and is the range the xy plane?

7. Originally Posted by cliggax
is the null space of the D transformation you calculated any point on the z axis? and is the range the xy plane?
The null space of the D transform would be any element of the form

$a \cdot e_1 +0\cdot e_2 +....0 \cdot e_{k+1}$

The null space is the set of constant polynomials. It is everything that is transformed into the Zero Vector

The range would be the span of the column Vectors

The column vectors span the subspace $P_k$ of $P_{k+1}$

8. is the function f(x) = 2x^2 +3x - 1 an arbitrary function? Essentially by multiplying

-1
3
2

by the other matrix are you finding the D transform of the function 2x^2 +3x - 1?

Could i substitute my own function in place of yours (of degree 2 of course)

9. Yes for example

$f(x)=22x^2-8x+7$ could be written as

$7 \cdot e_1 -8 \cdot e_2+ 22 \cdot e_3$ or in vector form

$7 \cdot \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}
-8 \cdot \begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}
+ 22 \cdot
\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}
=
\begin{bmatrix}
7 \\
-8 \\
22
\end{bmatrix}
$