/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q31Q Let \(T:{\mathbb{R}^n} \to {\mat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q31Q (page 1)

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, with A its standard matrix. Complete the following statement to make it true: 鈥�T is one-to-one if and only if A has 颅颅____ pivot columns.鈥� Explain why the statement is true. [Hint:Look in the exercises for Section 1.7.]

Short Answer

Expert verified

T is one-to-one if and only if A has 颅颅\(n\) pivot columns.

Step by step solution

01

Identify the condition for the transformation of dimensions

For matrix Aof order\(m \times n\), if the vector\({\bf{x}}\)is in\({\mathbb{R}^n}\), then transformation\(T\)of vector x is represented as\(T\left( x \right)\), and it is in the dimension\({\mathbb{R}^m}\).

It can also be written as\(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\).

Here, the dimension \({\mathbb{R}^n}\) is the domain, and the dimension \({\mathbb{R}^m}\) is the codomain of transformation \(T\).

02

Complete the statement to make it true

If the columns of A are linearly independent, then transformation T is one-to-one.

Matrix A has n pivot columns if they are linearly independent.

Thus, the correct statement is 鈥�T is one-to-one if and only if A has 颅颅\(n\) pivot columns.鈥�

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If Ais a 2脳2matrix with eigenvalues 3 and 4 and if localid="1668109698541" u鈫� is a unit eigenvector of A, then the length of vector Alocalid="1668109419151" u鈫�cannot exceed 4.

Consider a dynamical system x鈫�(t+1)=Ax鈫�(t) with two components. The accompanying sketch shows the initial state vector x鈫�0and two eigen vectors 蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫� of A (with eigen values 位1鈫�and位2鈫� respectively). For the given values of 位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=1,位2鈫�=0.9

In Exercises 11 and 12, determine if \({\rm{b}}\) is a linear combination of \({{\mathop{\rm a}\nolimits} _1},{a_2}\) and \({a_3}\).

11.\({a_1} = \left[ {\begin{array}{*{20}{c}}1\\{ - 2}\\0\end{array}} \right],{a_2} = \left[ {\begin{array}{*{20}{c}}0\\1\\2\end{array}} \right],{a_3} = \left[ {\begin{array}{*{20}{c}}5\\{ - 6}\\8\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\\6\end{array}} \right]\)

Write the reduced echelon form of a \(3 \times 3\) matrix A such that the first two columns of Aare pivot columns and

\(A = \left( {\begin{aligned}{*{20}{c}}3\\{ - 2}\\1\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}0\\0\\0\end{aligned}} \right)\).

Question: There exists a 2x2 matrix such thatA[12]=[34].
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.