Q3.3-74E Find a basis of the row space of... [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 3: Q3.3-74E (page 146)

Find a basis of the row space of the matrix
$A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 1 & 2 & 3 & 4 \\ 1 & 3 & 5 & 7 \end{matrix}]$

Short Answer

Expert verified

The basis of the row space of the given matrix is $\{[\begin{matrix} 1 & 0 & - 1 & - 2 \end{matrix}], [\begin{matrix} 0 & 1 & 2 & 3 \end{matrix}]\}$ .

Step by step solution

Definition of row space

The row space of a matrix is defined as the collection of all linear combinations of its row and the basis of the row space of a matrix is defined as the non-zero rows of the matrix which are reduced to row echelon form and are independent.

Given

We are given a matrix

$A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 1 & 2 & 3 & 4 \\ 1 & 3 & 5 & 7 \end{matrix}]$

Using row transformation

The row space is equal to the row space $r r e f (A)$ .

Using row transformations to find the basis of row space:

$R_{2} 鈫� R_{2} - 2 R_{1}$ and $R_{3} 鈫� R_{3} - R_{1}$

$r r e f (A) = [\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 3 \\ 1 & 3 & 5 & 7 \end{matrix}]$

$R_{4} 鈫� R_{4} - R_{1}$

$r r e f (A) = [\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 3 \\ 0 & 2 & 4 & 6 \end{matrix}]$

$R_{4} 鈫� R_{4} - 2 R_{3}$

localid="1664350039385" $r r e f (A) = [\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \end{matrix}]$

Replace R2 with R1

$R_{2} 鈫� R_{1}$ and $R_{1} 鈫� R_{1} - R_{3}$

$r r e f (A) = [\begin{matrix} 1 & 0 & - 1 & - 2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$

The final values

From here we can read the basis of the row space are $\{[\begin{matrix} 1 & 0 & - 1 & - 2 \end{matrix}], [\begin{matrix} 0 & 1 & 2 & 3 \end{matrix}]\}$ .

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91影视

Find a basis of the row space of the matrix
$A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 1 & 2 & 3 & 4 \\ 1 & 3 & 5 & 7 \end{matrix}]$

Short Answer

Step by step solution

Definition of row space

Given

Using row transformation

Replace R2 with R1

The final values

One App. One Place for Learning.

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91影视

Find a basis of the row space of the matrixA=[1111222212341357]

Short Answer

Step by step solution

Definition of row space

Given

Using row transformation

Replace R2 with R1

The final values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Applied Mathematics

Discrete Mathematics

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

Find a basis of the row space of the matrix
$A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 1 & 2 & 3 & 4 \\ 1 & 3 & 5 & 7 \end{matrix}]$