tamola7f
2022-09-13
Answered

Every matrix represents a linear transformation, but depending on characteristics of the matrix, the linear transformation it represents can be limited to a specific type. For example, an orthogonal matrix represents a rotation (and possibly a reflection). Is it something similar about triangular matrices? Do they represent any specific type of transformation?

You can still ask an expert for help

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 2021-09-15

Assume that T is a linear transformation. Find the standard matrix of T. $T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{4},T\left({e}_{1}\right)=(3,1,3,1)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\left({e}_{2}\right)=(-5,2,0,0),\text{}where\text{}{e}_{1}=(1,0)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{e}_{2}=(0,1)$ .

asked 2022-06-20

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

asked 2022-06-14

If $T:{\mathbb{R}}^{3}\to {P}^{2}$ where $T(a,b)=(a-b){t}^{2}+(-a+b)$, is $T$ a matrix transformation?

asked 2021-11-05

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

$$A=\left[\begin{array}{cccc}1& 3& 5& 0\\ 0& 1& 4& -2\end{array}\right]$$