Izabella Ponce

Answered

2022-06-26

If

$A=\left[\begin{array}{ccc}1& -1& 2\\ -2& 1& -1\\ 1& 2& 3\end{array}\right]$

is the matrix representation of a linear transformation

$T:{P}_{2}(x)\to {P}_{2}(x)$

with respect to the bases $\{1-x,x(1-x),x(1+x)\}$ and $\{1,1+x,1+{x}^{2}\}$ then find T. What is the procedure to solve it?

$A=\left[\begin{array}{ccc}1& -1& 2\\ -2& 1& -1\\ 1& 2& 3\end{array}\right]$

is the matrix representation of a linear transformation

$T:{P}_{2}(x)\to {P}_{2}(x)$

with respect to the bases $\{1-x,x(1-x),x(1+x)\}$ and $\{1,1+x,1+{x}^{2}\}$ then find T. What is the procedure to solve it?

Answer & Explanation

crociandomh

Expert

2022-06-27Added 19 answers

Recall the the matrix of a transformation has as its columns the images of the domain basis vectors expressed relative to a basis of the codomain. Assuming that the first of the given bases is for the domain, this means that

$T[1-x]=1\cdot (1)-2\cdot (1+x)+1\cdot (1+{x}^{2})={x}^{2}-2x.$

The other two columns give you $T[x(1-x)]$ and $T[x(1+x)]$, respectively. Now, express the general second-degree polynomial $a+bx+c{x}^{2}$ as a linear combination of the domain basis polynomials, i.e., as $\alpha (1-x)+\beta x(1-x)+\gamma x(1+x)$ and use linearity of $T$:

$T[a+bx+cy]=\alpha T[1-x]+\beta T[x(1-x)]+\gamma T[x(1+x)].$$T[a+bx+cy]=\alpha T[1-x]+\beta T[x(1-x)]+\gamma T[x(1+x)].$

$T[1-x]=1\cdot (1)-2\cdot (1+x)+1\cdot (1+{x}^{2})={x}^{2}-2x.$

The other two columns give you $T[x(1-x)]$ and $T[x(1+x)]$, respectively. Now, express the general second-degree polynomial $a+bx+c{x}^{2}$ as a linear combination of the domain basis polynomials, i.e., as $\alpha (1-x)+\beta x(1-x)+\gamma x(1+x)$ and use linearity of $T$:

$T[a+bx+cy]=\alpha T[1-x]+\beta T[x(1-x)]+\gamma T[x(1+x)].$$T[a+bx+cy]=\alpha T[1-x]+\beta T[x(1-x)]+\gamma T[x(1+x)].$

Eden Solomon

Expert

2022-06-28Added 7 answers

The columns of the matrix of a transformation are the images of the basis vectors of the domain expressed relative to the basis of the codomain, i.e., it specifies a linear combination of the codomain basis vectors. So, the first column of A tells us that $T[1-x]=1\cdot (1)-2\cdot (1+x)+1\cdot (1+{x}^{2})={x}^{2}-2x$, and so on. From that, you should be able to work out what T does to the general polynomial $a+bx+c{x}^{2}$. Alternatively, you might convert A to the standard basis and read the solution from that.

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