Answer to Question #254802 in Linear Algebra for Sabelo Xulu

Question

4.a) Let R[x] deg≤2 be the vector space of all polynomial functions in a single variable x with real coefficient and degree at most 2

Let R[x] deg≤2$\to$R[x] deg≤2 be defined by T(p(x)) =p(x-1)

Let U = <1,x,x²> be the usual basis for R[x] deg≤2 and B = <1+x+x²,2x+x²,x+x²> be another basis for R[x] deg≤2

Let, for any polynomial p(x) $\isin$ R[x] deg≤2, coord_B(p(x)) R³ is the coordinates of p(x) with respect to the basis B for instance

coord_V(x+x²) =(0, 1,1)

and

coord_B(x+x²)=(0, 0,1)

Also recall the agreement every vector X=(x₁, x₂, x₃) $\isin$ R³ is considered as a 3x1 matrix $\begin{pmatrix} X1 \\ X2 \\ X3 \end{pmatrix}$ i.e, as a column vector

a) Find the matrix Mat_U$\to$ _U (T) representing the linear transformation T with respect to the basis of U

b) Verify the equation Mat_U$\to$ _U (T)coord_U(p(x)) =coord_U(T(p(x)), for any p(x) $\isin$ R[x] deg≤2

c) Obtain the change of basis matrix P_B$\to$_U from B to U

d) Hance, or otherwise, obtain coord_B(p(x))for each p(x) $\isin$ R[x] deg≤2

Answer 1

a) $T((P(x))=P(x-1)$

$T(1)=0 , T(x)=x-1,$

$T(x)$ ² $=x$ ²$-1$

With respect to U

$0= a(1)+b(x)+c(x$ ²$)$

$a=0\:,b=0,c=0 ,$

$[0]_u =[0,0,0]^T$

$x-1=a(1)+b(x)+c(x^2)$

$a=-1,b=1,c=0$

$[x-1]_u=[-1,1,0]^T$

$x^2-1=a(1)+b(x)+c(x^2)$

$a=-1,b=0,c=1$

$[x^2-1]_u=[-1,0,1]^T$

Matrix $_U$ $\to$ _u(T)$=\begin{pmatrix} 0&-1 & -1\\ 0&1 & 0\\ 0&0&1 \end{pmatrix}$

Part B

Coord_u$(P(x))= \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$

Coord _u$(T(P(x)))=$

Coord_u $(P(x-1))=\begin{bmatrix} -1 \\ 1\\ 0 \end{bmatrix}$

$\begin{pmatrix} 0&-1 & -1\\ 0&1 & 0\\ 0&0&1 \end{pmatrix}$ $\begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix}$ =$\begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix}$

Verified

Part C

Expressing elements of B in terms of the elements of U

$1+x+x^2=a(1)+b(x)+c(x^2)$

$a=1,b=1,c=1$

$[1+x+x^2]_u=[1,1,1]^T$

$2x+x^2=a(1)+b(x)+c(x^2)$

$a=0,b=2,c=1$

$[2x+x^2]_u=[0,2,1]^T$

$x+x^2=a(1)+b(x)+c(x^2)$

$a=0,b=1,c=1$

$[x+x^2]_u=[0,1,1]^T$

Change of base matrix $P_B\to_U=$

$\begin{bmatrix} 1&0 & 0\\ 1&2 & 1\\ 1&1&1 \end{bmatrix}$

Part D

Expressing elements of Coord_u $(P(x))$ in terms of basis B

$x=a(1+x+x^2)+b(2x+x^2)$

$+c(x+x^2)$

$x=a+(a+2b+c)x+(a+b+c)x^2$

$\implies a=0,a+2b+c=1$

$\implies 2b+c=1.......(i)$

$a+b+c=0$

$\implies b+c=0......(ii)$

Solving $(i)$ and$(ii)$

$b=1,c=-1$

Coord$_B(P(x))=\begin{pmatrix} 0 \\ 1\\ -1 \end{pmatrix}$

Alternatively (method 2)

Coord_u $(P(x))=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$

Change of basis matrix $P_u\to_B=$

$(Matrix P_B\to_u)^{-1}$

$\therefore \begin{pmatrix} 1&0 & 0 \\ 1&2 & 1\\ 1&1&1 \end{pmatrix}$ $^-1$ ( coord$_B(P(x))$)

$\begin{pmatrix} 1&0& 0\\ 0&1&-1\\ -1&-1&2 \end{pmatrix}$ $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ =$\begin{pmatrix} 0\\ 1\\ -1 \end{pmatrix}$