o solve the following **Title:** Finding a Basis and Dimension of the Solution Space for a Homogeneous System of Linear Equations---**Problem Statement:**Find a basis for and the dimension of the solution space of the homogeneous system of linear equations.\[\begin{align*}- x_1 + 2x_2 - x_3 + 2x_4 &= 0 \\-2x_1 + 2x_2 + x_3 + 4x_4 &= 0 \\3x_1 + 2x_2 + 2x_3 + 5x_4 &= 0 \\-9x_1 + 14x_2 + 8x_3 + 29x_4 &= 0\end{align*}\]---**(a) A Basis for the Solution Space:**The diagram is a column vector representing a basis for the solution space:\[\begin{bmatrix}0 \\-3/2 \\-1 \\1 \\\end{bmatrix}\]- The vector indicates a key linear combination that satisfies the homogeneous system, forming a basis.---**(b) The Dimension of the Solution Space:**- The user-input field shows the number "2", marked with a red cross, indicating that the dimension stated is incorrect.---By solving the system using methods such as row reduction, one can determine the correct basis and dimension of the solution space. ### Matrix Determinant and Verification Problem#### TaskFind the determinants $|A|$, $|B|$, and $|AB|$. Then verify that $|A||B| = |AB|$.#### Given MatricesMatrix $A$:\[A = \begin{bmatrix} 0 & 1 & 2 \\ 5 & 4 & 3 \\ 7 & 6 & 8 \end{bmatrix}\]Matrix $B$:\[B = \begin{bmatrix} 5 & 1 & 5 \\ 1 & -1 & 0 \\ 0 & 4 & -5 \end{bmatrix}\]#### Steps to Solve(a) Calculate $|A|$: The determinant of matrix $A$.(b) Calculate $|B|$: The determinant of matrix $B$.(c) Calculate the product $AB$: First, find the product of matrices $A$ and $B$. This will result in a new matrix, denoted as $AB$.(d) Calculate $|AB|$: Determine the determinant of the resulting matrix $AB$.#### VerificationVerify that the equation $|A||B| = |AB|$ holds true with the calculated determinants. #### Diagram ExplanationThe diagram includes placeholders for calculations with boxes representing the elements of the resulting matrix $AB$. Each placeholder needs to be filled with the computed values from the matrix multiplication.- **Arrows** indicate the process of calculating each element of $AB$ by using the dot product of corresponding rows of $A$ with columns of $B$.- **Green arrows** (down and across) guide which rows and columns should interact.Fill in the matrix product $AB$ using the elements derived from these calculations before determining $|AB|$.This exercise involves matrix multiplication, calculation of determinants, and verification of determinant properties in linear algebra.

"Hey, since there are multiple questions posted, we will answer the first question. If you want any specific question to be answered then specify the question number in your message". The matrix form for homogeneous linear system is: -12-12-22143225-914829x1x2x3x4=0000 Find the row-echelon form: -12-12-22143225-914829=-914829-22143225-12-12R1↔R4 =-9148290-109-79-2293225-12-12R2←R2-29R1 =-9148290-109-79-2290203143443-12-12R3←R3+13R1 =-9148290-109-79-2290203143443049-179-119R4←R4-19R1 =-91482902031434430-109-79-229049-179-119R2↔R3 =-9148290203143443000000-115-115R3←R3+16R2 and R4←R4-115R2 =-914829020314344300-115-1150000R3↔R4 From the reduced row echelon form, we have: -9x1+14x2+8x3+29x4=0 ------(1)203x2+143x3+443x4=0 ------(2)-115x3-115x4=0 --------(3) where x4 is a free variable. Substitute x4 = t in equation (3): (3)⇒-115x3-115t=0 -115x3=115t x3 =-t Substitute x3=-t in equation (2): 2⇒203x2+143-t+443t=0 203x2=-10t x2 = -32t Substitute x4 = t , x3=-t and x2=-32t in equation (1): 1⇒-9x1+14-32t-8t+29t=0 -9x1-21t-8t+29t=0 -9x1=0 x1=0 Write in matrix form: x1x2x3x4=0-32t-tt=0-32-11t Therefore, a) The basis for the solution space is 0-32-11 b) Since, we have only one vector in the base. Therefore, Dimension of the solution space is 1.

Math

Algebra

Find a basis for and the dimension of the solution space of the homogeneous system of linear equations. -x1 + 2x2 - -2x1 + 2x2 + X3 + 4x4 = 0 X3 + 2x4 = 0 %3D 3x1 + 2x2 + 2x3 + 5x4 = 0 -9x1 + 14x2 + 8x3 + 29x4 = 0

Find a basis for and the dimension of the solution space of the homogeneous system of linear equations. -x1 + 2x2 - -2x1 + 2x2 + X3 + 4x4 = 0 X3 + 2x4 = 0 %3D 3x1 + 2x2 + 2x3 + 5x4 = 0 -9x1 + 14x2 + 8x3 + 29x4 = 0

Algebra and Trigonometry (6th Edition)

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ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

o solve the following

**Title:** Finding a Basis and Dimension of the Solution Space for a Homogeneous System of Linear Equations

---

**Problem Statement:**

Find a basis for and the dimension of the solution space of the homogeneous system of linear equations.

\[
\begin{align*}
- x_1 + 2x_2 - x_3 + 2x_4 &= 0 \\
-2x_1 + 2x_2 + x_3 + 4x_4 &= 0 \\
3x_1 + 2x_2 + 2x_3 + 5x_4 &= 0 \\
-9x_1 + 14x_2 + 8x_3 + 29x_4 &= 0
\end{align*}
\]

---

**(a) A Basis for the Solution Space:**

The diagram is a column vector representing a basis for the solution space:

\[
\begin{bmatrix}
0 \\
-3/2 \\
-1 \\
1 \\
\end{bmatrix}
\]

- The vector indicates a key linear combination that satisfies the homogeneous system, forming a basis.

---

**(b) The Dimension of the Solution Space:**

- The user-input field shows the number "2", marked with a red cross, indicating that the dimension stated is incorrect.

---

By solving the system using methods such as row reduction, one can determine the correct basis and dimension of the solution space.

### Matrix Determinant and Verification Problem

#### Task
Find the determinants $|A|$, $|B|$, and $|AB|$. Then verify that $|A||B| = |AB|$.

#### Given Matrices
Matrix $A$:
\[
A = \begin{bmatrix}
0 & 1 & 2 \\
5 & 4 & 3 \\
7 & 6 & 8
\end{bmatrix}
\]

Matrix $B$:
\[
B = \begin{bmatrix}
5 & 1 & 5 \\
1 & -1 & 0 \\
0 & 4 & -5
\end{bmatrix}
\]

#### Steps to Solve

(a) Calculate $|A|$:
The determinant of matrix $A$.

(b) Calculate $|B|$:
The determinant of matrix $B$.

(c) Calculate the product $AB$:
First, find the product of matrices $A$ and $B$. This will result in a new matrix, denoted as $AB$.

(d) Calculate $|AB|$:
Determine the determinant of the resulting matrix $AB$.

#### Verification
Verify that the equation $|A||B| = |AB|$ holds true with the calculated determinants.

#### Diagram Explanation
The diagram includes placeholders for calculations with boxes representing the elements of the resulting matrix $AB$. Each placeholder needs to be filled with the computed values from the matrix multiplication.

- **Arrows** indicate the process of calculating each element of $AB$ by using the dot product of corresponding rows of $A$ with columns of $B$.

- **Green arrows** (down and across) guide which rows and columns should interact.

Fill in the matrix product $AB$ using the elements derived from these calculations before determining $|AB|$.

This exercise involves matrix multiplication, calculation of determinants, and verification of determinant properties in linear algebra.

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