**Vectors as Linear Combinations***Problem Statement:*Write each vector as a linear combination of the vectors in the set $ S $. Use $ s_1 $ and $ s_2 $ for the vectors in the set $ S $. If not possible, enter "IMPOSSIBLE."*Given Set:*\[ S = \{ (6, -7, 8, 6), (4, 6, -4, 1) \} \]**Vectors:**(a) \[ \mathbf{u} = (-54, 127, -128, -72) \]Linear combination form: \[ \mathbf{u} = \_\_\_\_\](b)\[ \mathbf{v} = \left( \frac{41}{2}, \frac{75}{4}, -10, \frac{17}{2} \right) \]Linear combination form: \[ \mathbf{v} = \_\_\_\_\](c)\[ \mathbf{w} = \left( -2, -11, \frac{23}{8}, \frac{57}{8} \right) \]Linear combination form: \[ \mathbf{w} = \_\_\_\_\](d)\[ \mathbf{z} = \left( 4, -2, 3, \frac{13}{4} \right) \]Linear combination form: \[ \mathbf{z} = \_\_\_\_\]*Instructions:*To express each vector as a linear combination of the vectors in the set $ S $, solve for coefficients $ a $ and $ b $ such that:\[ a \cdot (6, -7, 8, 6) + b \cdot (4, 6, -4, 1) = \text{[given vector]} \] If it is not possible to form a linear combination for a given vector, write "IMPOSSIBLE".

Answered: Image /qna-images/answer/4f318cbb-e02f-48ac-96d0-61ce39f16a83.jpg

Write each vector as a linear combination of the vectors in S. (Use s and s, respectively, for the vectors in the set. If not possible, enter IMP S- ((6, -7, 8, 6), (4, 6,-4, 1)} (a) u-(-54, 127, -128,-72) -(4-10,). (b) (c) W3D

Write each vector as a linear combination of the vectors in S. (Use s and s, respectively, for the vectors in the set. If not possible, enter IMP S- ((6, -7, 8, 6), (4, 6,-4, 1)} (a) u-(-54, 127, -128,-72) -(4-10,). (b) (c) W3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Magnitude

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.

Question

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**Vectors as Linear Combinations**

*Problem Statement:*
Write each vector as a linear combination of the vectors in the set $ S $. Use $ s_1 $ and $ s_2 $ for the vectors in the set $ S $. If not possible, enter "IMPOSSIBLE."

*Given Set:*
\[ S = \{ (6, -7, 8, 6), (4, 6, -4, 1) \} \]

**Vectors:**

(a)
\[
\mathbf{u} = (-54, 127, -128, -72)
\]
Linear combination form:
\[
\mathbf{u} = \_\_\_\_
\]

(b)
\[
\mathbf{v} = \left( \frac{41}{2}, \frac{75}{4}, -10, \frac{17}{2} \right)
\]
Linear combination form:
\[
\mathbf{v} = \_\_\_\_
\]

(c)
\[
\mathbf{w} = \left( -2, -11, \frac{23}{8}, \frac{57}{8} \right)
\]
Linear combination form:
\[
\mathbf{w} = \_\_\_\_
\]

(d)
\[
\mathbf{z} = \left( 4, -2, 3, \frac{13}{4} \right)
\]
Linear combination form:
\[
\mathbf{z} = \_\_\_\_
\]

*Instructions:*

To express each vector as a linear combination of the vectors in the set $ S $, solve for coefficients $ a $ and $ b $ such that:
\[
a \cdot (6, -7, 8, 6) + b \cdot (4, 6, -4, 1) = \text{[given vector]}
\]

If it is not possible to form a linear combination for a given vector, write "IMPOSSIBLE".

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