$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)$

Feinsn
2022-06-26
Answered

Suppose $T$ is an element of $L({P}_{3}(\mathbb{R}),{P}_{2}(\mathbb{R}))$ is the differentiation map defined by $Tp={p}^{\prime}$. Find a basis of ${P}_{3}(\mathbb{R})$ and a basis of ${P}_{2}(\mathbb{R})$ such that the matrix of T with respect to these basis is

$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)$

$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)$

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asked 2021-09-18

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

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 2021-09-13

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

asked 2020-11-24

Let n be a fixed positive integer greater thatn 1 and let a and b be positive integers. Prove that a mod n = b mon n if and only if a = b mod.

asked 2022-04-12

For the displacement of the element by n rows from the identity matrix, the element should hold the value $(\frac{1}{2}{)}^{n}$ in the transformed matrix.

Here are a few examples:

$A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\to {A}^{\prime}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

$B=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)\to {B}^{\prime}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& \frac{1}{2}\\ 0& \frac{1}{2}& 0\end{array}\right)$

$C=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)\to {C}^{\prime}=\left(\begin{array}{ccc}0& \frac{1}{2}& 0\\ 0& 0& \frac{1}{2}\\ \frac{1}{4}& 0& 0\end{array}\right)$

How can such a transformation be realized mathematically?

Here are a few examples:

$A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\to {A}^{\prime}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

$B=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)\to {B}^{\prime}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& \frac{1}{2}\\ 0& \frac{1}{2}& 0\end{array}\right)$

$C=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)\to {C}^{\prime}=\left(\begin{array}{ccc}0& \frac{1}{2}& 0\\ 0& 0& \frac{1}{2}\\ \frac{1}{4}& 0& 0\end{array}\right)$

How can such a transformation be realized mathematically?

asked 2022-05-17

I have to find components of a matrix for 3D transformation. I have a first system in which transformations are made by multiplying:

${M}_{1}=[Translation]\times [Rotation]\times [Scale]$

I want to have the same transformations in an engine who compute like this:

${M}_{2}=[Rotation]\times [Translation]\times [Scale]$

So when I enter the same values there's a problem due to the inversion of translation and rotation.

How can I compute the values in the last matrix ${M}_{2}$ for having the same transformation?

${M}_{1}=[Translation]\times [Rotation]\times [Scale]$

I want to have the same transformations in an engine who compute like this:

${M}_{2}=[Rotation]\times [Translation]\times [Scale]$

So when I enter the same values there's a problem due to the inversion of translation and rotation.

How can I compute the values in the last matrix ${M}_{2}$ for having the same transformation?

asked 2022-06-13

Given $c$ in $R$, define ${T}_{c}:{R}^{n}\to R$ by ${T}_{c}(x)=cx$ for all $x$ in ${R}^{n}$. Show that ${T}_{c}$ is a linear transformation and find its matrix.

I don't understand the question. For ${T}_{c}$ is $c$ the matrix and we are supposed to find $c$? Would the matrix just be

$\begin{array}{ccc}c& 0& 0...0\end{array}$

I don't understand the question. For ${T}_{c}$ is $c$ the matrix and we are supposed to find $c$? Would the matrix just be

$\begin{array}{ccc}c& 0& 0...0\end{array}$