Q29E Find the basis of all聽2脳2 matr... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q29E (page 176)

Find the basis of all $2 脳 2$ matrixA such that $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ , and determine its dimension.

Short Answer

Expert verified

The dimension of a basis ofA is 2 which is spanned by $S p a n \{[\begin{matrix} 1 & - 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & - 1 \end{matrix}]\}$ .

Step by step solution

Determine the matrix .

Consider the equation $A [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ where $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ .

Substitute the value $[\begin{matrix} a & b \\ c & d \end{matrix}]$ for A in the equation $A [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ as follows.

$A [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} a + b & a + b \\ c + d & c + d \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

Compare the equation both side as follows.

$\begin{array}{l} a + b = 0 \\ c + d = 0 \end{array}$

Simplify the equation $a + b = 0$ as follows.

localid="1659414423987" $\begin{array}{l} a + b = 0 \\ a = - b \end{array}$

Simplify the equation $c + d = 0$ as follows.

$c + d = 0 c = - d$

Substitute the values $- c$ for d and -a for b in the equation $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ as follows.

$\begin{array}{l} A = [\begin{matrix} a & b \\ c & d \end{matrix}] \\ A = [\begin{matrix} a & - a \\ c & - c \end{matrix}] \end{array}$

Therefore, the matrix $A = [\begin{matrix} a & - a \\ c & - c \end{matrix}]$ .

Determine the basis of the matrix .

Simplify the equation $A = [\begin{matrix} a & - a \\ c & - c \end{matrix}]$ as follows.

$A = [\begin{matrix} a & - a \\ c & - c \end{matrix}] A = [\begin{matrix} a & - a \\ c & - c \end{matrix}] + [\begin{matrix} 0 & 0 \\ 1 & - 1 \end{matrix}] A = a [\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix}] + c [\begin{matrix} 0 & 0 \\ 1 & - 1 \end{matrix}]$

Therefore, the matrix A is spanned by $S p a n \{[\begin{matrix} 1 & - 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & - 1 \end{matrix}]\}$ where $[\begin{matrix} 1 & - 1 \\ 0 & 0 \end{matrix}]$ and $[\begin{matrix} 0 & 0 \\ 1 & - 1 \end{matrix}]$ are linear independent.

Hence, the dimension ofA is 2 and spanned by $S p a n \{[\begin{matrix} 1 & - 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & - 1 \end{matrix}]\}$ .

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the basis of all $2 脳 2$ matrixA such that $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ , and determine its dimension.

Short Answer

Step by step solution

Determine the matrix .

Determine the basis of the matrix .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Statistics

Theoretical and Mathematical Physics

Geometry

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

91影视

Find the basis of all2脳2 matrixA such thatA=[1101]=[0000], and determine its dimension.

Short Answer

Step by step solution

Determine the matrix .

Determine the basis of the matrix .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Statistics

Theoretical and Mathematical Physics

Geometry

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Find the basis of all $2 脳 2$ matrixA such that $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ , and determine its dimension.