linear algebra **Problem 1**: What vectors \( \mathbf{b} \) are in the column space of \( \mathbf{A} \)? Give constraint equations. (Reference: ch3.2.2a)Matrix \( \mathbf{A} \) is given by:\[\mathbf{A} = \begin{bmatrix} 3 & -1 \\ 6 & -2 \\ -9 & 3 \end{bmatrix}\]To determine which vectors \( \mathbf{b} \) are in the column space of \( \mathbf{A} \), we must express \( \mathbf{b} \) as a linear combination of the columns of \( \mathbf{A} \). The column space of \( \mathbf{A} \) is the set of all vectors that can be written as:\[c_1 \begin{bmatrix} 3 \\ 6 \\ -9 \end{bmatrix} + c_2 \begin{bmatrix} -1 \\ -2 \\ 3 \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}\]Here, \( c_1 \) and \( c_2 \) are scalars, and \( \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \) is the resulting vector \( \mathbf{b} \). Vectors \( \mathbf{b} \) in the column space must satisfy this form.

To find The vector b are in the column space of A.

Answered: 1. What vectors b are in the column…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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linear algebra

**Problem 1**: What vectors \( \mathbf{b} \) are in the column space of \( \mathbf{A} \)? Give constraint equations. (Reference: ch3.2.2a)

Matrix \( \mathbf{A} \) is given by:

\[
\mathbf{A} = \begin{bmatrix} 3 & -1 \\ 6 & -2 \\ -9 & 3 \end{bmatrix}
\]

To determine which vectors \( \mathbf{b} \) are in the column space of \( \mathbf{A} \), we must express \( \mathbf{b} \) as a linear combination of the columns of \( \mathbf{A} \). The column space of \( \mathbf{A} \) is the set of all vectors that can be written as:

\[
c_1 \begin{bmatrix} 3 \\ 6 \\ -9 \end{bmatrix} + c_2 \begin{bmatrix} -1 \\ -2 \\ 3 \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}
\]

Here, \( c_1 \) and \( c_2 \) are scalars, and \( \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \) is the resulting vector \( \mathbf{b} \). Vectors \( \mathbf{b} \) in the column space must satisfy this form.

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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