### Solving Linear Systems Using LU Factorization#### Problem StatementSolve the linear system $ \mathbf{A}\mathbf{x} = \mathbf{b} $ by finding an $ LU $-factorization of the coefficient matrix $\mathbf{A}$, solving the lower triangular system $ \mathbf{L}\mathbf{y} = \mathbf{b} $, and solving the upper triangular system $ \mathbf{U}\mathbf{x} = \mathbf{y} $.Given system:\[ \begin{aligned}2x + y &= 1 \\y - z &= 2 \\-2x + y + z &= -2 \\\end{aligned}\]#### Steps to Solve**(a) Find an $ LU $-factorization of the coefficient matrix $\mathbf{A}$.** Your $ \mathbf{L} $ matrix must be unit diagonal.The process involves decomposing matrix $\mathbf{A}$ into a lower triangular matrix $\mathbf{L}$ and an upper triangular matrix $\mathbf{U}$.\[LU = \begin{bmatrix} & & \\ & & \\ & & \end{bmatrix}\rightarrow\begin{bmatrix} & & \\ & & \\ & & \end{bmatrix}\rightarrow\begin{bmatrix} & & \\ & & \\ & & \end{bmatrix}\]**(b) Solve the lower triangular system $ \mathbf{L}\mathbf{y} = \mathbf{b} $.** \[y = \begin{bmatrix} \\ \\ \end{bmatrix}\rightarrow\]**(c) Solve the upper triangular system $ \mathbf{U}\mathbf{x} = \mathbf{y} $.** \[x = \begin{bmatrix} \\ \\ \end{bmatrix}\rightarrow\]### Diagram Explanation1. **LU Decomposition:** - **Matrix $\mathbf{L}$**: A 3x3 lower triangular matrix with a unit diagonal (diagonal elements are 1). - **Matrix $\mathbf{U}$**: A 3x3 upper

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Solve the linear system Ax = b by finding an LU-factorization of the coefficient matrix A, solving the lower triangular system Ly = b, and solving the upper triangular system Ux = y. 2x + y 1 %3! y - z = -2x + y + z = -2 (a) Find an LU-factorization of the coefficient matrix A. (Your L matrix must be unit diagonal.) LU = (b) Solve the lower triangular system Ly = b. y = (c) Solve the upper triangular system Ux = y. X = D00

Solve the linear system Ax = b by finding an LU-factorization of the coefficient matrix A, solving the lower triangular system Ly = b, and solving the upper triangular system Ux = y. 2x + y 1 %3! y - z = -2x + y + z = -2 (a) Find an LU-factorization of the coefficient matrix A. (Your L matrix must be unit diagonal.) LU = (b) Solve the lower triangular system Ly = b. y = (c) Solve the upper triangular system Ux = y. X = D00

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Solving Linear Systems Using LU Factorization

#### Problem Statement
Solve the linear system $ \mathbf{A}\mathbf{x} = \mathbf{b} $ by finding an $ LU $-factorization of the coefficient matrix $\mathbf{A}$, solving the lower triangular system $ \mathbf{L}\mathbf{y} = \mathbf{b} $, and solving the upper triangular system $ \mathbf{U}\mathbf{x} = \mathbf{y} $.

Given system:
\[
\begin{aligned}
2x + y &= 1 \\
y - z &= 2 \\
-2x + y + z &= -2 \\
\end{aligned}
\]

#### Steps to Solve

**(a) Find an $ LU $-factorization of the coefficient matrix $\mathbf{A}$.**

Your $ \mathbf{L} $ matrix must be unit diagonal.

The process involves decomposing matrix $\mathbf{A}$ into a lower triangular matrix $\mathbf{L}$ and an upper triangular matrix $\mathbf{U}$.

\[
LU =
\begin{bmatrix}
& & \\
& & \\
& &
\end{bmatrix}
\rightarrow
\begin{bmatrix}
& & \\
& & \\
& &
\end{bmatrix}
\rightarrow
\begin{bmatrix}
& & \\
& & \\
& &
\end{bmatrix}
\]

**(b) Solve the lower triangular system $ \mathbf{L}\mathbf{y} = \mathbf{b} $.**

\[
y =
\begin{bmatrix}
\\
\\

\end{bmatrix}
\rightarrow
\]

**(c) Solve the upper triangular system $ \mathbf{U}\mathbf{x} = \mathbf{y} $.**

\[
x =
\begin{bmatrix}
\\
\\

\end{bmatrix}
\rightarrow
\]

### Diagram Explanation

1. **LU Decomposition:**
- **Matrix $\mathbf{L}$**: A 3x3 lower triangular matrix with a unit diagonal (diagonal elements are 1).
- **Matrix $\mathbf{U}$**: A 3x3 upper

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