Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q63E (page 58)

Solving a linear system $A \overset{鈫�}{x} = \overset{鈫�}{0}$ by Gaussian elimination amounts to writing the vector of leading variables as a linear transformation of the vector of free variables. Consider the linear system.
$\begin{matrix} x_{1} 鈭� x_{2} + 4 x_{5} = 0 \\ x_{3} 鈭� x_{5} = 0 \\ x_{4} 鈭� 2 x_{5} = 0 \end{matrix}$
Find the matrix Bsuch that $[\begin{array}{l} x_{1} \\ x_{3} \\ x_{4} \end{array}] = B [\begin{array}{l} x_{2} \\ x_{5} \end{array}]$ .

Short Answer

Expert verified

The matrix, $B = [\begin{matrix} 1 & 鈭� 4 \\ \begin{array}{l} 0 \\ 0 \end{array} & \begin{array}{l} 1 \\ 2 \end{array} \end{matrix}]$ .

Step by step solution

Consider the linear system

The system of equations is,

$\begin{matrix} x_{1} 鈭� x_{2} + 4 x_{5} = 0 \\ x_{3} 鈭� x_{5} = 0 \\ x_{4} 鈭� 2 x_{5} = 0 \end{matrix}$

The matrix form of the equations is,

$[\begin{matrix} 1 & 鈭� 1 & 0 & 0 & 4 & 0 \\ 0 & 0 & 1 & 0 & 鈭� 1 & 0 \\ 0 & 0 & 0 & 1 & 鈭� 2 & 0 \end{matrix}]$

Compute the matrix.

The row-echelon form of the matrix is,

$[\begin{matrix} 1 & 鈭� 1 & 0 & 0 & 4 & 0 \\ 0 & 0 & 1 & 0 & 鈭� 1 & 0 \\ 0 & 0 & 0 & 1 & 鈭� 2 & 0 \end{matrix}]$

Find the matrix

Represent the equation in terms of a matrix.

$\begin{matrix} x_{1} 鈭� x_{2} + 4 x_{5} = 0 \\ 鈬� x_{1} = x_{2} 鈭� 4 x_{5} \\ x_{3} 鈭� x_{5} = 0 \\ 鈬� x_{3} = x_{5} \\ x_{4} 鈭� 2 x_{5} = 0 \\ 鈬� x_{4} = 2 x_{5} \end{matrix}$

The matrix is,

role="math" localid="1664194616664" $\begin{matrix} [\begin{array}{l} x_{1} \\ x_{3} \\ x_{4} \end{array}] = B [\begin{array}{l} x_{2} \\ x_{5} \end{array}] \\ 鈬� [[\begin{array}{l} x_{1} \\ x_{3} \\ x_{4} \end{array}]] = [\begin{matrix} 1 & 鈭� 4 \\ \begin{array}{l} 0 \\ 0 \end{array} & \begin{array}{l} 1 \\ 2 \end{array} \end{matrix}] [\begin{array}{l} x_{2} \\ x_{5} \end{array}] \\ 鈭� B = [\begin{matrix} 1 & 鈭� 4 \\ \begin{array}{l} 0 \\ 0 \end{array} & \begin{array}{l} 1 \\ 2 \end{array} \end{matrix}] \end{matrix}$

Hence, the matrix, $B = [\begin{matrix} 1 & 鈭� 4 \\ \begin{array}{l} 0 \\ 0 \end{array} & \begin{array}{l} 1 \\ 2 \end{array} \end{matrix}]$ such that $[\begin{array}{l} x_{1} \\ x_{3} \\ x_{4} \end{array}] = B [\begin{array}{l} x_{2} \\ x_{5} \end{array}]$ .

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Short Answer

Step by step solution

Consider the linear system

Compute the matrix.

Find the matrix

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91影视

Solving a linear system Ax鈫�=0鈫�by Gaussian elimination amounts to writing the vector of leading variables as a linear transformation of the vector of free variables. Consider the linear system.x1鈭�x2+4x5=0x3鈭�x5=0x4鈭�2x5=0Find the matrix Bsuch that[x1x3x4]=B[x2x5].

Short Answer

Step by step solution

Consider the linear system

Compute the matrix.

Find the matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Logic and Functions

Theoretical and Mathematical Physics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.