Linear Algebra. **Problem Statement:**The vector $ \mathbf{x} $ is in a subspace $ H $ with a basis $ B = \{\mathbf{b}_1, \mathbf{b}_2\} $. Find the $ B $-coordinate vector of $ \mathbf{x} $.**Given:**\[\mathbf{b}_1 = \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \quad \mathbf{b}_2 = \begin{bmatrix} -1 \\ 3 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} 4 \\ 9 \end{bmatrix}\]**Solution Framework:**To find the $ B $-coordinate vector of $ \mathbf{x} $, denoted as $ [\mathbf{x}]_B $, we express $ \mathbf{x} $ as a linear combination of the basis vectors $ \mathbf{b}_1 $ and $ \mathbf{b}_2 $.Thus, we seek scalars $ c_1 $ and $ c_2 $ such that:\[c_1 \begin{bmatrix} 2 \\ 1 \end{bmatrix} + c_2 \begin{bmatrix} -1 \\ 3 \end{bmatrix} = \begin{bmatrix} 4 \\ 9 \end{bmatrix}\]This leads to the system of equations:1. $ 2c_1 - c_2 = 4 $2. $ c_1 + 3c_2 = 9 $Solving this system will give the values for $ c_1 $ and $ c_2 $.**[x]_B = [ ? ]**Calculate the corresponding $ B $-coordinates to fill in the blank.

We have given the basis B=b1,b2where b1=21 , b2=-13 and we have a vector x = 49in a subspace H basis B.To find the B-coordinate vector ofx we have to write x as a linearcombination of the vectors b1 and b2Let x =λ1b1 + λ2b2 , where λ1 , λ2 are real scalars not all zero.Now 49 =λ1 21 + λ2-13 or 49 = 2λ1-λ2λ1+3λ2 Comparing the like elements from both sides we get,2λ1-λ2=4 , or , 2λ1-λ2-4=0λ1+3λ2=9 , or , λ1+3λ1-9=0By cross-multiplication we get, λ19+12=λ2-4+18=16+1or, λ121=λ214=17∴λ121=17 , or , λ1=217=3and λ214=17 , or , λ2=147=2∴ the B-coordinate vector of x =xB=λ1λ2 = 32

Answered: The vector x is in a subspace H with a…

Algebra and Trigonometry (6th Edition)

6th Edition

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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Question

Linear Algebra.

**Problem Statement:**

The vector $ \mathbf{x} $ is in a subspace $ H $ with a basis $ B = \{\mathbf{b}_1, \mathbf{b}_2\} $. Find the $ B $-coordinate vector of $ \mathbf{x} $.

**Given:**

\[
\mathbf{b}_1 = \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \quad \mathbf{b}_2 = \begin{bmatrix} -1 \\ 3 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} 4 \\ 9 \end{bmatrix}
\]

**Solution Framework:**

To find the $ B $-coordinate vector of $ \mathbf{x} $, denoted as $ [\mathbf{x}]_B $, we express $ \mathbf{x} $ as a linear combination of the basis vectors $ \mathbf{b}_1 $ and $ \mathbf{b}_2 $.

Thus, we seek scalars $ c_1 $ and $ c_2 $ such that:

\[
c_1 \begin{bmatrix} 2 \\ 1 \end{bmatrix} + c_2 \begin{bmatrix} -1 \\ 3 \end{bmatrix} = \begin{bmatrix} 4 \\ 9 \end{bmatrix}
\]

This leads to the system of equations:

1. $ 2c_1 - c_2 = 4 $
2. $ c_1 + 3c_2 = 9 $

Solving this system will give the values for $ c_1 $ and $ c_2 $.

**[x]_B = [ ? ]**

Calculate the corresponding $ B $-coordinates to fill in the blank.

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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