W₁ = B W2 = (9) where a ER, BER. (a) Give an expression for ww2 (b) Does the linear independence of these vectors depend on 3? (c) Are there any values of a for which these vectors form a basis?
W₁ = B W2 = (9) where a ER, BER. (a) Give an expression for ww2 (b) Does the linear independence of these vectors depend on 3? (c) Are there any values of a for which these vectors form a basis?
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
Consider two vectors:
![The given text involves two vectors defined as follows:
\[ w_1 = \beta \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \quad w_2 = \begin{pmatrix} \alpha \\ -1 \end{pmatrix} \]
where \(\alpha \in \mathbb{R}\) and \(\beta \in \mathbb{R}\).
The questions posed are:
(a) Give an expression for \(w_1^T w_2\).
(b) Does the linear independence of these vectors depend on \(\beta\)?
(c) Are there any values of \(\alpha\) for which these vectors form a basis?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1da88d38-e18c-40f1-9bb8-69bad3269ec7%2F190f60aa-bbf6-4a13-826e-ebcc19a85fed%2Fdad75t1_processed.png&w=3840&q=75)
Transcribed Image Text:The given text involves two vectors defined as follows:
\[ w_1 = \beta \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \quad w_2 = \begin{pmatrix} \alpha \\ -1 \end{pmatrix} \]
where \(\alpha \in \mathbb{R}\) and \(\beta \in \mathbb{R}\).
The questions posed are:
(a) Give an expression for \(w_1^T w_2\).
(b) Does the linear independence of these vectors depend on \(\beta\)?
(c) Are there any values of \(\alpha\) for which these vectors form a basis?
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