Ytukyg

I'm currently studying for a Linear Algebra exam in january, thus I'm going through some older exam questions, and I'm at the following question.

Let $mathbb{R}[x]$ denote the set of all real polynomials, which is organized as a vectorspace over $mathbb{R}$.

• A. Assume that $U={pinmathbb{R}_{1}[x]|p(2)=0}$. Determine a basis for $U$ and its dimension.


• B. Add $p(x)=x-2$ to a basis $B$ for $mathbb{R}_3[x]$, and determine the coordinate vector $[q]_B$ for $q(x)=3+2x+x^2$
  


• C. Determine a subspace $Wsubset mathbb{R}_3[x]$ such that $Uoplus W=mathbb{R}_3[x]$. Determine $dim W$.


• D. Find the matrix $M(T)$ with respect to $B$ when $T=4D+3I$, where $D$ denote differentiation and $I$ denotes the identical map. Determine whether $T$ is injective, surjective or bijective.


• E. Let $L$ denote the set of all real numbers $lambda$ such that $S=4D-lambda I$ has $span(1,x^2,x^3)$ as a subspace which is invariant under $S$. Determine $L$.

My answers are,

A. $q(x)=x-2$ is a basis for $U$ and its dimension is 1.

B. I add $p(x)=x-2$ to the standard basis of $mathbb{R}_3[x]$, such that $B=(x-2,1,x,x^2,x^3)$, this is not a basis of $mathbb{R}_3[x]$ as $x$ can be written as a linear comb. of the previous elements (vectors?), thus I reduce it to a basis by removing $x$ from the list so $B=(x-2,1,x^2,x^3)$. Now I write the matrix with respect to this basis and add $begin{bmatrix}3\2\1\0end{bmatrix}$ to the matrix, and then I compute the coordinate vector $[q]_B$ by gauss elimination.

$begin{align}
begin{bmatrix}
-2 & 1 & 0 & 0 & 3\
1 & 0 & 0 & 0 & 2\
0 & 0 & 1 & 0& 1\
0 & 0 & 0 & 1 & 0
end{bmatrix}
sim
begin{bmatrix}
1 & 0 & 0 & 0 & 2\
0 & 1 & 0 & 0 & 7\
0 & 0 & 1 & 0 & 1\
0 & 0 & 0 & 1 & 0
end{bmatrix}
end{align}$

Thus
$[q]_B = begin{bmatrix}2 \7\1\0end{bmatrix}$

C. From B. $mathbb{R}_3[x]$ had the basis $(x-2,1,x^2,x^3)$, and the basis of $U=(x-2)$, so $W=(1,x^2,x^3)$, thus $Uoplus W$ will span $mathbb{R}_3[x]$. $dim W=3$

I'm stuck at D. and E. I assume that I start by differentiating the basis of $B$, then representing $B'$ as a matrix which I substitute into $D$ and then add the $3I$? I'm not sure that I'm capable of solving E. without knowing the methods from D.

I hope that some of it is right, and I hope to get some tips, correction and some help with the last two questions.

Best regards Jens.

Up till now all correct :)
For exercise D) you'll need The differentiation Matrix $$D=
begin{bmatrix}
0 & 1 & 0 & 0 \
0 & 0 & 2 & 0 \
0 & 0 & 0 & 3 \
0 & 0 & 0 & 0
end{bmatrix}$$
$$T:=4D+3I=begin{bmatrix}
3 & 4 & 0 & 0 \
0 & 3 & 8 & 0 \
0 & 0& 3 & 12 \
0 & 0 & 0 & 3
end{bmatrix} $$
T is bijective because $det(T)=3^4neq 0$ If you have not learned about determinants jet you can simply use the Gauss-algorithm to Calculate $A^{-1}$.
Then have to write T with as a Matrix with Respect to the Basis B. For this you will need the Transformation Matricies $[id]^B_S$ and $[id]^S_B$ (where S denotes the standart-Basis) then $[T]^B_B=[id]^B_S T [id]^S_B$

hint:
$[id]^B_S$ the transformation matrix from S to B is:
$$[id]^B_S=begin{bmatrix}
-2 & 1 & 0 & 0 \
1 & 0 & 0 & 0 \
0 & 0 & 1& 0 \
0 & 0 & 0 & 1
end{bmatrix} $$
to understand this look at $[id]^B_S e_j $.

Also note $[id]^B_S=([id]^S_B)^{-1}$