Linear algebra number 16 1.4 ### Understanding Solutions of Linear Systems#### 15. Analyzing Solutions for Specific Matrices and VectorsLet $ A = \begin{bmatrix}3 & -1 \\9 & 3 \end{bmatrix} $ and $ b = \begin{bmatrix}b_1 \\b_2 \end{bmatrix} $. Show that the equation $ Ax = b $ does not have a solution for all possible $ b $, and describe the set of all $ b $ for which $ Ax = b $ **does** have a solution.#### 16. Repetition with a Different Set of Matrices and VectorsRepeat the requests from Exercise 15 with the following $ A $ and $ b $:\[ A = \begin{bmatrix}1 & 2 & -1 \\2 & 2 & 0 \\4 & 1 & 3 \end{bmatrix}, \quad b = \begin{bmatrix}b_1 \\b_2 \\b_3 \end{bmatrix} \]#### 17-20. Further Exploration with Matrices $ A $ and $ B $Exercises 17-20 refer to the matrices $ A $ and $ B $ below. Make appropriate calculations that justify your answers and mention an appropriate theorem.\[ A = \begin{bmatrix}1 & 3 & 0 & 3 \\1 & 1 & 1 & 0 \\0 & 4 & 2 & 8 \\2 & 0 & 1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix}1 & 4 & 2 \\0 & 1 & 4 \\0 & 2 & 6 \\2 & 0 & 7 \end{bmatrix} \]##### 17. Pivot Positions in $ A $How many rows of $ A $ contain a pivot position? Does the equation $ Ax = b $ have a solution for each $ b $ in $ \mathbb{R}^4 $?##### 18. Linear Combinations and Spanning in $ \mathbb{R}^4 $Can every vector in $ \mathbb{R}^4 $

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Answered: 15. Let A and b Show that the cquation…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Linear algebra number 16 1.4

### Understanding Solutions of Linear Systems

#### 15. Analyzing Solutions for Specific Matrices and Vectors

Let $ A = \begin{bmatrix}
3 & -1 \\
9 & 3
\end{bmatrix} $ and $ b = \begin{bmatrix}
b_1 \\
b_2
\end{bmatrix} $. Show that the equation $ Ax = b $ does not have a solution for all possible $ b $, and describe the set of all $ b $ for which $ Ax = b $ **does** have a solution.

#### 16. Repetition with a Different Set of Matrices and Vectors

Repeat the requests from Exercise 15 with the following $ A $ and $ b $:

\[
A = \begin{bmatrix}
1 & 2 & -1 \\
2 & 2 & 0 \\
4 & 1 & 3
\end{bmatrix}, \quad
b = \begin{bmatrix}
b_1 \\
b_2 \\
b_3
\end{bmatrix}
\]

#### 17-20. Further Exploration with Matrices $ A $ and $ B $

Exercises 17-20 refer to the matrices $ A $ and $ B $ below. Make appropriate calculations that justify your answers and mention an appropriate theorem.

\[
A = \begin{bmatrix}
1 & 3 & 0 & 3 \\
1 & 1 & 1 & 0 \\
0 & 4 & 2 & 8 \\
2 & 0 & 1 & 3
\end{bmatrix}, \quad
B = \begin{bmatrix}
1 & 4 & 2 \\
0 & 1 & 4 \\
0 & 2 & 6 \\
2 & 0 & 7
\end{bmatrix}
\]

##### 17. Pivot Positions in $ A $

How many rows of $ A $ contain a pivot position? Does the equation $ Ax = b $ have a solution for each $ b $ in $ \mathbb{R}^4 $?

##### 18. Linear Combinations and Spanning in $ \mathbb{R}^4 $

Can every vector in $ \mathbb{R}^4 $

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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