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

Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a linear transformation, with $A$ itsstandard matrix. Complete the following statement to make it true: $T$ is one-to-one if and only if $A$ has $-$ pivot columns." Explain why the statement is true. [Hint: Look in the exercises for Section 1.7 .1

$T$ is one-to-one if and only if $A$ has $n$ pivot columns.

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 9

The Matrix of a Linear Transformation

Introduction to Matrices

Missouri State University

Baylor University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

01:00

Let $T : \mathbb{R}^{n} \r…

02:44

a. Fill in the blank in th…

01:58

Let $\mathbf{A}$ be a matr…

00:54

In Exercises $1-10$ , assu…

01:15

Determine the linear trans…

00:52

00:53

02:52

Verify the uniqueness of $…

04:42

In Exercises 17 and 18 , $…

in this video, we're handling a transformation. T mapping are in into RM which can be written in this form. T F X equals eight times X because we have a standard matrix here. This implies already that t is linear, which is a very helpful piece of information. Tohave. Now that we know that T is linear, we can ascertain the following that t is one, 21 if and only if all use the bi directional arrows for the F in Olean Statement T or not T but rather the standard matrix A has a pivot in every call him, So it's describe why we know there must be a pivot in every column in order for tea to be oneto one. Well, we have a theory that says the following t is one 21 If and only if a has Lee Nearly independent columns. Let's say what a having linearly Independent columns is equivalent to best game. It happens if and only if The matrix equation eight times X equals zero has no free variables. Or, in other words, a X equals zero has on Lee the trivial solution. But we know that that can happen if and only if The Matrix A has a pivot in every column. So that's all right. Our ideas tied together to give this equivalent statement.

View More Answers From This Book

Find Another Textbook

Numerade Educator

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

Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a linear transformati…

a. Fill in the blank in the following statement: "If $A$ is an $m \time…

Let $\mathbf{A}$ be a matrix. Show that the matrix $\mathbf{A} \mathbf{A}^{t…

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

Determine the linear transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R…

Verify the uniqueness of $A$ in Theorem $10 .$ Let $T : \mathbb{R}^{n} \righ…

In Exercises 17 and 18 , $A$ is an $m \times n$ matrix. Mark each statement …

02:49

The given matrix determines a linear transformation $T .$ Find all $\mathbf{…

03:10

Write a matrix equation that determines the loop currents. [M] If MATLAB or …

01:13

In Exercises 1–9, assume that the matrices are partitioned conformably for b…

05:09

Use the algorithm from this section to find the inverses of $$\left[…

04:10

In Exercises $7-12$ , describe all solutions of $A \mathbf{x}=0$ in parametr…

03:09

Suppose $A$ is an $m \times n$ matrix and there exist $n \times m$ matrices …

02:10

An affine transformation $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ has…

01:09

How many rows does $B$ have if $B C$ is a $3 \times 4$ matrix?

03:07

05:37

Suppose the block matrix $A$ on the left side of (7) is invertible and $A_{1…

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.