**Vector Combinations in $ \mathbb{R}^3 $**Let's consider the set $ S = \{(1, 0, 1), (0, 1, 0)\} $ consisting of two vectors in $ \mathbb{R}^3 $. Our goal is to express the following vectors as linear combinations of the vectors in $ S $. If it is not feasible to do so, we will show the matrix that explains why it is not possible.(a) $(-4, 6, -4)$(b) $(-4, 3, 3)$(c) $(0, 24, 0)$(d) $(1, 0, 0)$**Explanation:**We aim to solve for constants $ a $ and $ b $ such that each of the given vectors can be written in the form:\[ a(1, 0, 1) + b(0, 1, 0) = (a, b, a) \]For each vector:1. Determine $ a $ and $ b $ if possible.2. Present the matrix representation that confirms whether the solutions exist.This structured approach helps in verifying and understanding the linear dependence of vectors in $ \mathbb{R}^3 $.

Answered: Image /qna-images/answer/bb5120a7-9a34-4a0f-a2c0-76ffed4e2d3a.jpg

Let S = {(1,0, 1), (0, 1,0)} be two vectors in R³. Write the following vectors as a linear combination of the vectors in S if possible. If it's not possible, show the matrix indicating why it isn't. (a) (-4, 6, –4) (b) (-4,3, 3) (c) (0,24, 0) (d) (1,0,0)

Let S = {(1,0, 1), (0, 1,0)} be two vectors in R³. Write the following vectors as a linear combination of the vectors in S if possible. If it's not possible, show the matrix indicating why it isn't. (a) (-4, 6, –4) (b) (-4,3, 3) (c) (0,24, 0) (d) (1,0,0)

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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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

**Vector Combinations in $ \mathbb{R}^3 $**

Let's consider the set $ S = \{(1, 0, 1), (0, 1, 0)\} $ consisting of two vectors in $ \mathbb{R}^3 $. Our goal is to express the following vectors as linear combinations of the vectors in $ S $. If it is not feasible to do so, we will show the matrix that explains why it is not possible.

(a) $(-4, 6, -4)$

(b) $(-4, 3, 3)$

(c) $(0, 24, 0)$

(d) $(1, 0, 0)$

**Explanation:**

We aim to solve for constants $ a $ and $ b $ such that each of the given vectors can be written in the form:

\[ a(1, 0, 1) + b(0, 1, 0) = (a, b, a) \]

For each vector:

1. Determine $ a $ and $ b $ if possible.
2. Present the matrix representation that confirms whether the solutions exist.

This structured approach helps in verifying and understanding the linear dependence of vectors in $ \mathbb{R}^3 $.

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