Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

In Exercises $17-20,$ show that $T$ is a linear transformation by finding a matrix that implements the mapping. Note that $x_{1}, x_{2}, \ldots$ are not vectors but are entries in vectors. $$T\left(x_{1}, x_{2}\right)=\left(2 x_{2}-3 x_{1}, x_{1}-4 x_{2}, 0, x_{2}\right)$$

$T(\mathbf{x})=\left[\begin{array}{cc}{-3} & {2} \\ {1} & {-4} \\ {0} & {0} \\ {0} & {1}\end{array}\right]\left[\begin{array}{l}{x_{1}} \\ {x_{2}}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 9

The Matrix of a Linear Transformation

Introduction to Matrices

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

03:45

In Exercises $17-20,$ show…

01:48

02:39

02:08

Let $\mathbf{x}=\left[\beg…

03:26

Column vectors are written…

01:38

In Exercises $1-10$ , assu…

02:41

Let $T_{1}: \mathbb{R}^{2}…

01:19

03:41

Assume that $T$ defines a …

So we want to, uh, find a vectors. We'll call it a such that when we multiply it by ex one next to it. Maps it too. This vector here on the right. So we want this to be written as a X to show that it's ah, linear transformation. So we have a matrix A and we don't necessarily know the size yet. We'll figure that out. No, we want to multiply it by x one x two and we want to get a vector with first component two x to minus three x one x one minus four x two zero in x two. So since our two component factor used to be mapped to a four component vector, this needs to be a four by two matrix. So we don't usually draw lines like Listen, Major sees. But I'm gonna do it just so we know how many spots we have the fill in. So we know that this whole first column is what's multiplying the X one and I could call on his multiplying the X to coordinate. So to get this to x two minus three x one x one has to be multiplied by negative three and X two has to be multiplied by two For the second component, X one has to be one supplied by one. And then we have minus four times x two. So you 20 We need to multiply both components by zero to ensure that we always have a zero in this third spot. Lastly, Thio, just get the output output. The second component, Um, our vector x for this fourth row here, we need to multiply X to buy one, and we need to get rid of X one. So we're going to multiply that by zero. So this here is our major X a.

View More Answers From This Book

Find Another Textbook

In mathematics, the absolute value or modulus |x| of a real number x is its …

In Exercises $17-20,$ show that $T$ is a linear transformation by finding a …

Let $\mathbf{x}=\left[\begin{array}{c}{x_{1}} \\ {x_{2}}\end{array}\right], …

Column vectors are written as rows, such as $\mathbf{x}=\left(x_{1}, x_{2}\r…

In Exercises $1-10$ , assume that $T$ is a linear transformation. Find the s…

Let $T_{1}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ and $T_{2}: \mathbb{R…

Assume that $T$ defines a linear transformation and use the given informatio…

01:37

Find formulas for X, Y, and Z in terms of A, B, and C, and justify your calc…

02:00

Let $A$ be an invertible $n \times n$ matrix, and let $B$ be an $n \times p$…

01:15

Unless otherwise specified, assume that all matrices in these exercises are …

05:20

A steam plant burns two types of coal: anthracite (A) and bituminous (B). Fo…

02:43

Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linea…

00:46

In Exercises 7–10, the augmented matrix of a linear system has been reduced …

Suppose the coefficient matrix of a system of linear equationshas a pivo…

15:25

A thin triangular plate of uniform density and thickness hasvertices at …

02:51

Let $A=\left[\begin{array}{rrr}{-2} & {-7} & {-9} \\ {2} & {5} &…

02:56

Let $\mathbf{u}=\left[\begin{array}{l}{7} \\ {2} \\ {5}\end{array}\right], \…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.