I took the standard basis for grade-n polynomials:

$B=<1,x,{x}^{2},\dots ,{x}^{n}>,\phi (B)=\phi (1,x,{x}^{2},...,{x}^{n})=(0,2x,...,n{x}^{n-1})$

So, the matrix is $\left[\begin{array}{c}0\\ 2x\\ \dots \\ n{x}^{n-1}\end{array}\right]$?

hawatajwizp
2022-06-20
Answered

Let $\phi :K[x{]}_{\le n}\to K[x{]}_{\le n-1}$ with $\phi $ the linear transformation defind by $\phi (f)={f}^{\prime}$. Select a base and find the matrix of the linear transformation.

I took the standard basis for grade-n polynomials:

$B=<1,x,{x}^{2},\dots ,{x}^{n}>,\phi (B)=\phi (1,x,{x}^{2},...,{x}^{n})=(0,2x,...,n{x}^{n-1})$

So, the matrix is $\left[\begin{array}{c}0\\ 2x\\ \dots \\ n{x}^{n-1}\end{array}\right]$?

I took the standard basis for grade-n polynomials:

$B=<1,x,{x}^{2},\dots ,{x}^{n}>,\phi (B)=\phi (1,x,{x}^{2},...,{x}^{n})=(0,2x,...,n{x}^{n-1})$

So, the matrix is $\left[\begin{array}{c}0\\ 2x\\ \dots \\ n{x}^{n-1}\end{array}\right]$?

You can still ask an expert for help

Ryan Newman

Answered 2022-06-21
Author has **26** answers

Since $\phi (1)=0$, $\phi (x)=1$, $\phi ({x}^{2})=2x$, and so on, the matrix is$\left[\begin{array}{ccccccc}0& 1& 0& 0& \dots & 0& 0\\ 0& 0& 2& 0& \dots & 0& 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0& 0& 0& 0& \dots & 0& n\end{array}\right].$Note that this matrix has $n(=\mathrm{dim}K[x{]}_{\le n-1})$ lines and $n+1(=\mathrm{dim}K[x{]}_{\le n})$ columns.

Jasmin Pineda

Answered 2022-06-22
Author has **2** answers

I don't really understand why it's constructed like this. From the 2nd column forward every pivot of that submatrix is the coefficient of each polynomial bu, I don't get why the first column is like so.

asked 2021-06-13

For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A.

$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$

Find a nonzero vector in Nul A.

$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

Find a nonzero vector in Nul A.

asked 2021-09-18

Find an explicit description of Nul A by listing vectors that span the null space.

asked 2021-09-13

Assume that A is row equivalent to B. Find bases for Nul A and Col A.

asked 2022-06-05

Similarity transformation of an orthogonal matrix

A transformation $T$ represented by an orthogonal matrix $A$ , so ${A}^{T}A=I$. This transformation leaves norm unchanged.

I do a basis change using a matrix $B$ which isn't orthogonal , then the form of the transformation changes to ${B}^{-1}AB$ in the new basis( A similarity transformation).

Therefore ${B}^{-1}AB$. $[{B}^{-1}AB{]}^{T}=I$.

This suggests that ${B}^{T}B=I$ which means it is orthogonal, but that is a contradiction.

A transformation $T$ represented by an orthogonal matrix $A$ , so ${A}^{T}A=I$. This transformation leaves norm unchanged.

I do a basis change using a matrix $B$ which isn't orthogonal , then the form of the transformation changes to ${B}^{-1}AB$ in the new basis( A similarity transformation).

Therefore ${B}^{-1}AB$. $[{B}^{-1}AB{]}^{T}=I$.

This suggests that ${B}^{T}B=I$ which means it is orthogonal, but that is a contradiction.

asked 2021-11-20

(a) find the transition matrix from B to $B}^{\prime$ ,

(b) find the transition matrix from$B}^{\prime$ to B,

(c) verify that the two transition matriced are inverses of each other, and

(d) find the coordinate matrix$\left[x\right]}_{B$ , given the coordinate matrix $\left[x\right]}_{B$ . $B=\{(1,3),(-2,-2)\},B\u2018=\{(-12,0),(-4,4)\}$

$${\left[x\right]}_{{B}^{\prime}}=\left[\begin{array}{c}-1\\ 3\end{array}\right]$$

(b) find the transition matrix from

(c) verify that the two transition matriced are inverses of each other, and

(d) find the coordinate matrix

asked 2021-12-13

Let ${a}_{1}=\left[\begin{array}{c}1\\ 3\\ -1\end{array}\right],{a}_{2}=\left[\begin{array}{c}-6\\ -14\\ 3\end{array}\right],b=\left[\begin{array}{c}5\\ 3\\ h\end{array}\right]$ . for what value(s) of h is b in the plane spanned by $a}_{1$ and $a}_{2$

asked 2021-08-08

Show that the system (G,⋅) is a group.