Write each vector as a linear combination of the vectors in S. (If not possible, enter IMPOSSIBLE.) S = {(2, –1, 3), (5, 0, 4)} (a) z = (1, -3, 5) S2 Z = 75 (b) 23, 4' 4 V = S2 V = (c) w = (6, -8, 16) %3D S2 W = (d) u = (9, 1, –1) S2 u = +
Write each vector as a linear combination of the vectors in S. (If not possible, enter IMPOSSIBLE.) S = {(2, –1, 3), (5, 0, 4)} (a) z = (1, -3, 5) S2 Z = 75 (b) 23, 4' 4 V = S2 V = (c) w = (6, -8, 16) %3D S2 W = (d) u = (9, 1, –1) S2 u = +
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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Question
![**Transcription for Educational Website:**
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**Write each vector as a linear combination of the vectors in \( S \). (If not possible, enter IMPOSSIBLE.)**
\( S = \{(2, -1, 3), (5, 0, 4)\} \)
---
**(a)** \( \mathbf{z} = (1, -3, 5) \)
\[
\mathbf{z} = \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_1 + \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_2
\]
**(b)** \( \mathbf{v} = \left( 23, -\frac{1}{4}, \frac{75}{4} \right) \)
\[
\mathbf{v} = \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_1 + \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_2
\]
**(c)** \( \mathbf{w} = (6, -8, 16) \)
\[
\mathbf{w} = \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_1 + \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_2
\]
**(d)** \( \mathbf{u} = (9, 1, -1) \)
\[
\mathbf{u} = \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_1 + \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_2
\]
---
**Explanation:**
This exercise involves expressing given vectors as linear combinations of two vectors in the set \( S \). Each of the unknown coefficients for \( \mathbf{s}_1 \) and \( \mathbf{s}_2 \) is to be determined or stated as "IMPOSSIBLE" if the vector cannot be represented as such a combination.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8268c271-8991-49a2-aed9-a02bde5bd8ab%2Fb2d533ff-f927-47b7-8488-2a1cd9e2eb77%2Fxrkhuvo_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
---
**Write each vector as a linear combination of the vectors in \( S \). (If not possible, enter IMPOSSIBLE.)**
\( S = \{(2, -1, 3), (5, 0, 4)\} \)
---
**(a)** \( \mathbf{z} = (1, -3, 5) \)
\[
\mathbf{z} = \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_1 + \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_2
\]
**(b)** \( \mathbf{v} = \left( 23, -\frac{1}{4}, \frac{75}{4} \right) \)
\[
\mathbf{v} = \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_1 + \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_2
\]
**(c)** \( \mathbf{w} = (6, -8, 16) \)
\[
\mathbf{w} = \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_1 + \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_2
\]
**(d)** \( \mathbf{u} = (9, 1, -1) \)
\[
\mathbf{u} = \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_1 + \left( \, \boxed{\phantom{0}} \, \right) \mathbf{s}_2
\]
---
**Explanation:**
This exercise involves expressing given vectors as linear combinations of two vectors in the set \( S \). Each of the unknown coefficients for \( \mathbf{s}_1 \) and \( \mathbf{s}_2 \) is to be determined or stated as "IMPOSSIBLE" if the vector cannot be represented as such a combination.
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