Q5E Let聽 be the solution of the lin... [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 5: Q5E (page 245)

Let be the solution of the linear system
$| \begin{matrix} x_{1} + x_{2} + x_{3} + x_{4} = 0 \\ x_{1} + 2 x_{2} + 5 x_{3} + 4 x_{4} = 0 \end{matrix} |$ .
Find the basis of $V^{鈯�}$ .

Short Answer

Expert verified

The set $\{{\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{1}, . . . ., {\overset{鈫�}{v}}_{p}, {\overset{鈫�}{w}}_{1}, {\overset{鈫�}{w}}_{1}, . . . {\overset{鈫�}{w}}_{q}\}$ spanned ${鈩�}^{n}$ .

Step by step solution

Determine the Matrix.

Consider the linear system $| \begin{matrix} x_{1} + x_{2} + x_{3} + x_{4} = 0 \\ x_{1} + 2 x_{2} + 5 x_{3} + 4 x_{4} = 0 \end{matrix} |$ .

Simplify $| \begin{matrix} x_{1} + x_{2} + x_{3} + x_{4} = 0 \\ x_{1} + 2 x_{2} + 5 x_{3} + 4 x_{4} = 0 \end{matrix} |$ as follows.

$| \begin{matrix} x_{1} + x_{2} + x_{3} + x_{4} = 0 \\ x_{1} + 2 x_{2} + 5 x_{3} + 4 x_{4} = 0 \end{matrix} | [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 5 & 4 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = 0$

Assume that $A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 5 & 4 \end{matrix}]$ .

Determine the kernel.

As the set V is set of all solution of the linear system implies $V = k e r (A)$ .

Assume $B = \{{\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{1}, . . . ., {\overset{鈫�}{v}}_{p}, {\overset{鈫�}{w}}_{1}, {\overset{鈫�}{w}}_{1}, . . . {\overset{鈫�}{w}}_{q}\}$ and $\overset{鈫�}{v} 鈭� {鈩�}^{n}$ .

Theorem: For any matrix B, $i m {(B)}^{鈯�} = k e r B^{T}$

By the theorem $i m {(A^{T})}^{鈯�} = k e r A^{T^{T}}$

As $A^{T^{T}} = A$ and $A^{{鈯�}^{鈯�}} = A$ , take $鈯�$ both side in the equation $i m {(A^{T})}^{鈯�} = k e r A^{T^{T}}$ as follows.

$i m {(A^{T})}^{鈯�} = k e r A^{T^{T}} i m {(A^{T})}^{鈯�} = k e r A {\{i m {(A^{T})}^{鈯�}\}}^{鈯�} = {(k e r A)}^{鈯�} i m {(A^{T})}^{鈯�} = {(k e r A)}^{鈯�}$

Therefore, the value $V^{鈯�} = i m (A^{T})$ .

Determine the matrix AT .

Simplify $A^{T}$ as follows.

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

By the definition of $i m (A)$ , the basis for $i m (A^{T})$ is $\{[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 5 \\ 4 \end{matrix}]\}$ .

Hence, the basis for $V^{鈯�}$ is $\{[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 5 \\ 4 \end{matrix}]\}$ .

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91影视

Let be the solution of the linear system
$| \begin{matrix} x_{1} + x_{2} + x_{3} + x_{4} = 0 \\ x_{1} + 2 x_{2} + 5 x_{3} + 4 x_{4} = 0 \end{matrix} |$ .
Find the basis of $V^{鈯�}$ .

Short Answer

Step by step solution

Determine the Matrix.

Determine the kernel.

Determine the matrix AT .

One App. One Place for Learning.

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91影视

Let be the solution of the linear system|x1+x2+x3+x4=0x1+2x2+5x3+4x4=0|.Find the basis ofV鈯�.

Short Answer

Step by step solution

Determine the Matrix.

Determine the kernel.

Determine the matrix AT .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Statistics

Probability and Statistics

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Let be the solution of the linear system
$| \begin{matrix} x_{1} + x_{2} + x_{3} + x_{4} = 0 \\ x_{1} + 2 x_{2} + 5 x_{3} + 4 x_{4} = 0 \end{matrix} |$ .
Find the basis of $V^{鈯�}$ .