9.2 If possible, find two vectors ū and v so that (a) a and c are non-negative linear combinations of u and but is not. (b) a and è are non-negative linear combinations of u and v. (c) a and are non-negative linear combinations of u and ✓ but d is not. (d) a, c, and à are convex linear combinations of u and v. Otherwise, explain why it's not possible.
9.2 If possible, find two vectors ū and v so that (a) a and c are non-negative linear combinations of u and but is not. (b) a and è are non-negative linear combinations of u and v. (c) a and are non-negative linear combinations of u and ✓ but d is not. (d) a, c, and à are convex linear combinations of u and v. Otherwise, explain why it's not possible.
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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need explanation on 9.2
Transcribed Image Text:**9.2** If possible, find two vectors **u** and **v** so that
(a) **a** and **c** are non-negative linear combinations of **u** and **v** but **b** is not.
(b) **a** and **e** are non-negative linear combinations of **u** and **v**.
(c) **a** and **b** are non-negative linear combinations of **u** and **v** but **d** is not.
(d) **a**, **c**, and **d** are convex linear combinations of **u** and **v**.
Otherwise, explain why it's not possible.
![On this educational page, we explore vector combinations. Consider the following vectors:
\[
\mathbf{a} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} -1 \\ 1 \end{bmatrix}, \quad \mathbf{c} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad \mathbf{d} = \begin{bmatrix} 0 \\ 2 \end{bmatrix}, \quad \mathbf{e} = \begin{bmatrix} -1 \\ -1 \end{bmatrix}.
\]
### 9.1 Analysis
Determine which of these vectors are:
(a) **Linear combinations** of \(\mathbf{a}\) and \(\mathbf{b}\)?
(b) **Non-negative linear combinations** of \(\mathbf{a}\) and \(\mathbf{b}\)?
(c) **Convex linear combinations** of \(\mathbf{a}\) and \(\mathbf{b}\)?
In the context of linear algebra, we explore various types of combinations to understand the relationships between these vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a8af915-3f7a-4a41-a3ab-e20554fd1f78%2F8a95044e-79c1-41de-b3e1-caacd6c1a682%2F8kh2zdm_processed.png&w=3840&q=75)
Transcribed Image Text:On this educational page, we explore vector combinations. Consider the following vectors:
\[
\mathbf{a} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} -1 \\ 1 \end{bmatrix}, \quad \mathbf{c} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad \mathbf{d} = \begin{bmatrix} 0 \\ 2 \end{bmatrix}, \quad \mathbf{e} = \begin{bmatrix} -1 \\ -1 \end{bmatrix}.
\]
### 9.1 Analysis
Determine which of these vectors are:
(a) **Linear combinations** of \(\mathbf{a}\) and \(\mathbf{b}\)?
(b) **Non-negative linear combinations** of \(\mathbf{a}\) and \(\mathbf{b}\)?
(c) **Convex linear combinations** of \(\mathbf{a}\) and \(\mathbf{b}\)?
In the context of linear algebra, we explore various types of combinations to understand the relationships between these vectors.
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