Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Question:A linear system of the form $A \vec{x} = 0$ **is called homogeneous. Justify the following facts:
a. All homogeneous systems are consistent.
b. A homogeneous system with fewer equations than unknowns has infinitely many solutions.
c. If $\vec{x_{1}} and \vec{x_{2}}$ are solutions of the homogeneous system $A \vec{x} = 0$ , then $\vec{x_{1}} + \vec{x_{2}}$ is a solution as well.
d. If $\vec{x}$ is a solution of the homogeneous system $A \vec{x} = 0$ and kis an arbitrary constant, then $k \vec{x}$ is a solution as well.**

a. $\vec{x} = 0$ is a solution for a linear system of the form and hence all the homogeneous systems are consistent.

b. Using theorem 1.3.3., homogeneous system with lesser equations than unknowns have infinitely many solutions.

c. Yes, $\vec{x_{1}} + \vec{x_{2}}$ is also a solution, if $\vec{x_{1}} a n d \vec{x_{2}}$ are solutions of the homogeneous system $A \vec{x} = 0$ .

d. Yes, $k \vec{x}$ is also a solution, if $\vec{x}$ is a solution of the homogeneous system $A \vec{x} = 0$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: (a) Consider the system.

If A is an $n \times m$ matrix with row vectors $\vec{ω 1}, . . ., \vec{ω n}$ and $\vec{x}$ is a vector in $ℝ^{m}$ then,

$\vec{A x} = [\begin{matrix} - {\vec{ω}}_{1} - \\ ⋮ \\ {\vec{ω}}_{n} - \end{matrix}] \vec{x} = [\begin{matrix} - {\vec{ω}}_{1} \cdot \vec{x} - \\ ⋮ \\ - {\vec{ω}}_{n} \cdot \vec{x} - \end{matrix}]$

Consider a linear system,

$A \vec{x} = \vec{b}$

Step 2: Determine the solution for the system.

When $\vec{b} = 0, A \vec{x} = 0$

Thus, $\vec{x} = 0$ will also be a solution for homogeneous system, $A \vec{x} = 0$ .

Step 3: (b) Consider a theorem

Theorem 1.3.3

a. For a linear system with at least one solution, then, the number of unknowns must be equal to the number of equations.

b. A linear system will have no solution or infinitely many solutions for number of equations less than the number of unknowns.

Considering the theorem 1.3.3, it can be stated that, a homogeneous system with fewer equations than unknowns has infinitely many solutions.

Step 4: (c) Consider the system.

Given:

$A \vec{x} = 0$ is a homogeneous linear system.

$\vec{x_{1}}$ and $\vec{x_{2}}$ are homogeneous solutions.

Consider the system,

$\begin{aligned} A \vec{x} = 0 \\ \Rightarrow A ({\vec{x}}_{1} + \vec{x_{2}}) = 0 \\ \Rightarrow A {\vec{x}}_{1} + A {\vec{x}}_{2} = 0 \\ ∴ {\vec{x}}_{1} + {\vec{x}}_{2} = 0 \end{aligned}$

Step 5:(d) Consider the system.

Given:

$A \vec{x} = 0$ is a homogeneous linear system.

$\vec{x}$ is a homogeneous solution.

k is a arbitrary constant.

Consider the system,

$\begin{aligned} A \vec{x} = 0 \\ \Rightarrow A (k \vec{x}) = 0 \\ \Rightarrow k (\vec{A x}) = 0 \\ \Rightarrow k (0) \vec{x} = 0 \\ ∴ \vec{k x} = 0 \end{aligned}$

Step 6: Final answer

a. $\vec{x} = 0$ is a solution.

b. Using theorem 1.3.3., homogeneous system with lesser equations than unknowns have infinitely many solutions.

c. Yes, $\vec{x_{1}} + \vec{x_{2}}$ is also a solution, if $\vec{x_{1}} and \vec{x_{2}}$ are solutions of the homogeneous system $A \vec{x} = 0$ .

d. Yes, $k \vec{x}$ is also a solution, if $\vec{x}$ is a solution of the homogeneous system $A \vec{x} = 0$ .

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Linear Algebra With Applications

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Step by Step Solution

Step 1: (a) Consider the system.

Step 2: Determine the solution for the system.

Step 3: (b) Consider a theorem

Step 4: (c) Consider the system.

Step 5:(d) Consider the system.

Step 6: Final answer

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

Step 1: (a) Consider the system.

Step 2: Determine the solution for the system.

Step 3: (b) Consider a theorem

Step 4: (c) Consider the system.

Step 5:(d) Consider the system.

Step 6: Final answer

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