The figure shows a basis B = {b, b2 } for R? and a vector v in R?. b1 b2 Custom basis B = {b1, b2} a. Write the vector v as a linear combination of the vectors in the basis B. Enter a vector sum the form 5 b1 +6 b2. v = Ob1+3b2 b. Find the B-coordinate vector for v. Enter your answer as a coordinate vector of the form <5,6>. [v]B = <0,3>
The figure shows a basis B = {b, b2 } for R? and a vector v in R?. b1 b2 Custom basis B = {b1, b2} a. Write the vector v as a linear combination of the vectors in the basis B. Enter a vector sum the form 5 b1 +6 b2. v = Ob1+3b2 b. Find the B-coordinate vector for v. Enter your answer as a coordinate vector of the form <5,6>. [v]B = <0,3>
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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![The figure shows a basis \( \mathcal{B} = \{ \mathbf{b_1}, \mathbf{b_2} \} \) for \( \mathbb{R}^2 \) and a vector \( \mathbf{v} \) in \( \mathbb{R}^2 \).
### Graph Description:
The graph displays a two-dimensional coordinate system with axes labeled \( x \) and \( y \). A custom basis \( \mathcal{B} = \{ \mathbf{b_1}, \mathbf{b_2} \} \) is shown, with basis vectors \( \mathbf{b_1} \) and \( \mathbf{b_2} \) represented as blue arrows originating from the origin. The vector \( \mathbf{v} \) is shown as a red arrow, indicating its position relative to the basis vectors.
### Tasks:
a. Write the vector \( \mathbf{v} \) as a linear combination of the vectors in the basis \( \mathcal{B} \). Enter a vector sum of the form \( 5 \mathbf{b_1} + 6 \mathbf{b_2} \).
- **Input Box:** \( \mathbf{v} = \) `0b1 + 3b2`
b. Find the \( \mathcal{B} \)-coordinate vector for \( \mathbf{v} \). Enter your answer as a coordinate vector of the form \( \langle 5, 6 \rangle \).
- **Input Box:** \( [\mathbf{v}]_{\mathcal{B}} = \) `<0, 3>`](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe82b7575-cd6e-46e0-8661-fced8d02a25c%2F452ea68c-9360-47cc-8189-ccc33792e911%2Fvnfoe56_processed.png&w=3840&q=75)
Transcribed Image Text:The figure shows a basis \( \mathcal{B} = \{ \mathbf{b_1}, \mathbf{b_2} \} \) for \( \mathbb{R}^2 \) and a vector \( \mathbf{v} \) in \( \mathbb{R}^2 \).
### Graph Description:
The graph displays a two-dimensional coordinate system with axes labeled \( x \) and \( y \). A custom basis \( \mathcal{B} = \{ \mathbf{b_1}, \mathbf{b_2} \} \) is shown, with basis vectors \( \mathbf{b_1} \) and \( \mathbf{b_2} \) represented as blue arrows originating from the origin. The vector \( \mathbf{v} \) is shown as a red arrow, indicating its position relative to the basis vectors.
### Tasks:
a. Write the vector \( \mathbf{v} \) as a linear combination of the vectors in the basis \( \mathcal{B} \). Enter a vector sum of the form \( 5 \mathbf{b_1} + 6 \mathbf{b_2} \).
- **Input Box:** \( \mathbf{v} = \) `0b1 + 3b2`
b. Find the \( \mathcal{B} \)-coordinate vector for \( \mathbf{v} \). Enter your answer as a coordinate vector of the form \( \langle 5, 6 \rangle \).
- **Input Box:** \( [\mathbf{v}]_{\mathcal{B}} = \) `<0, 3>`
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