Question

If T :mathbb{R}^2 rightarrow mathbb{R}^2 is a linear transformation such that Tleft(begin{bmatrix}3 6 end{bmatrix}right)=begin{bmatrix}39 33 end{bmatrix} text{ and } Tleft(begin{bmatrix}6 -5 end{bmatrix}right)=begin{bmatrix}27 -53 end{bmatrix} then the standard matrix of T is A

Matrices
ANSWERED
asked 2021-02-25
If T :\(\mathbb{R}^2 \rightarrow \mathbb{R}^2\) is a linear transformation such that
\(T\left(\begin{bmatrix}3 \\6 \end{bmatrix}\right)=\begin{bmatrix}39 \\33 \end{bmatrix} \text{ and } T\left(\begin{bmatrix}6 \\-5 \end{bmatrix}\right)=\begin{bmatrix}27 \\-53 \end{bmatrix}\)
then the standard matrix of T is A

Answers (1)

2021-02-26
Step 1
Given:
\(T\left(\begin{bmatrix}3 \\6 \end{bmatrix}\right)=\begin{bmatrix}39 \\33 \end{bmatrix} \text{ and } T\left(\begin{bmatrix}6 \\-5 \end{bmatrix}\right)=\begin{bmatrix}27 \\-53 \end{bmatrix}\)
The standard matrix of T will be the product of two matrices.
Step 2
\(T=\begin{pmatrix}3 & 39 \\6 & 33 \end{pmatrix}\begin{pmatrix}6 & 27 \\-5 & -53 \end{pmatrix}\)
\(=\begin{pmatrix}18-195 & 81-2067 \\36-165 & 162-1749 \end{pmatrix}\)
\(=\begin{pmatrix}-177 & -1986 \\-129 & -1587 \end{pmatrix}\)
Therefore the standard matrix of T is \(T = \begin{pmatrix}-177 & -1986 \\-129 & -1587 \end{pmatrix}\)
0
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours

Relevant Questions

asked 2020-10-20

In this problem, allow \(T_1: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) and \(T_2: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be linear transformations. Find \(Ker(T_1), Ker(T_2), Ker(T_3)\) of the respective matrices:
\(A=\begin{bmatrix}1 & -1 \\-2 & 0 \end{bmatrix} , B=\begin{bmatrix}1 & 5 \\-2 & 0 \end{bmatrix}\)

asked 2021-05-14
Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type.
\(\begin{array}{|c|c|}\hline 11.8 & 7.7 & 6.5 & 6 .8& 9.7 & 6.8 & 7.3 \\ \hline 7.9 & 9.7 & 8.7 & 8.1 & 8.5 & 6.3 & 7.0 \\ \hline 7.3 & 7.4 & 5.3 & 9.0 & 8.1 & 11.3 & 6.3 \\ \hline 7.2 & 7.7 & 7.8 & 11.6 & 10.7 & 7.0 \\ \hline \end{array}\)
a) Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion. \([Hint.\ ?x_{j}=219.5.]\) (Round your answer to three decimal places.)
MPa
State which estimator you used.
\(x\)
\(p?\)
\(\frac{s}{x}\)
\(s\)
\(\tilde{\chi}\)
b) Calculate a point estimate of the strength value that separates the weakest \(50\%\) of all such beams from the strongest \(50\%\).
MPa
State which estimator you used.
\(s\)
\(x\)
\(p?\)
\(\tilde{\chi}\)
\(\frac{s}{x}\)
c) Calculate a point estimate of the population standard deviation ?. \([Hint:\ ?x_{i}2 = 1859.53.]\) (Round your answer to three decimal places.)
MPa
Interpret this point estimate.
This estimate describes the linearity of the data.
This estimate describes the bias of the data.
This estimate describes the spread of the data.
This estimate describes the center of the data.
Which estimator did you use?
\(\tilde{\chi}\)
\(x\)
\(s\)
\(\frac{s}{x}\)
\(p?\)
d) Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 10 MPa. [Hint: Think of an observation as a "success" if it exceeds 10.] (Round your answer to three decimal places.)
e) Calculate a point estimate of the population coefficient of variation \(\frac{?}{?}\). (Round your answer to four decimal places.)
State which estimator you used.
\(p?\)
\(\tilde{\chi}\)
\(s\)
\(\frac{s}{x}\)
\(x\)
asked 2020-12-07

Let \(A=\begin{bmatrix}1 & 2 \\-1 & 1 \end{bmatrix} \text{ and } C=\begin{bmatrix}-1 & 1 \\2 & 1 \end{bmatrix}\)
a)Find elementary matrices \(E_1 \text{ and } E_2\) such that \(C=E_2E_1A\)
b)Show that is no elementary matrix E such that \(C=EA\)

asked 2020-12-28

Assume that T is a linear transformation. Find the standard matrix of T.
\(\displaystyle{T}=\mathbb{R}^{{2}}\rightarrow\mathbb{R}^{{4}}\ {s}{u}{c}{h} \ {t}hat \ {{T}}{\left({e}_{{1}}\right)}={\left({7},{1},{7},{1}\right)},\).

\({\quad\text{and}\quad}{T}{\left({e}_{{2}}\right)}={\left(-{8},{5},{0},{0}\right)},{w}{h}{e}{r}{e}{\ e}_{{1}}={\left({1},{0}\right)},\)

\({\quad\text{and}\quad}{e}_{{2}}={\left({0},{1}\right)}\)

...