1 +t 2 t 3+ 2t V = te R Define 1+t1 2 - t1 3+ 2t1 1+ t2 2 – t2 3+ 2t2 1+ (ti + t2) 2 (t1 + t2) 3+ 2(tı + t2) - and 1+ ct 2 - ct 1+t 2 t - 3+ 2t 3+ 2ct a. Find the additive identity and additive inverse. b. Show that V is a vector space (does it comply with ALL ten proper space?) Vor
1 +t 2 t 3+ 2t V = te R Define 1+t1 2 - t1 3+ 2t1 1+ t2 2 – t2 3+ 2t2 1+ (ti + t2) 2 (t1 + t2) 3+ 2(tı + t2) - and 1+ ct 2 - ct 1+t 2 t - 3+ 2t 3+ 2ct a. Find the additive identity and additive inverse. b. Show that V is a vector space (does it comply with ALL ten proper space?) Vor
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Vector Space Problem
1. Let
\[
V = \left\{ \begin{bmatrix} 1 + t \\ 2 - t \\ 3 + 2t \end{bmatrix} \mid t \in \mathbb{R} \right\}
\]
Define
\[
\begin{bmatrix}
1 + t_1 \\
2 - t_1 \\
3 + 2t_1
\end{bmatrix}
\oplus
\begin{bmatrix}
1 + t_2 \\
2 - t_2 \\
3 + 2t_2
\end{bmatrix}
=
\begin{bmatrix}
1 + (t_1 + t_2) \\
2 - (t_1 + t_2) \\
3 + 2(t_1 + t_2)
\end{bmatrix}
\]
and
\[
c \odot
\begin{bmatrix}
1 + t \\
2 - t \\
3 + 2t
\end{bmatrix}
=
\begin{bmatrix}
1 + ct \\
2 - ct \\
3 + 2ct
\end{bmatrix}
\]
#### Tasks:
a. Find the additive identity and additive inverse.
b. Show that \( V \) is a vector space (does it comply with ALL ten properties of a vector space?).
c. Verify that \( 0 \cdot \mathbf{v} = \mathbf{0} \) for all \(\mathbf{v}\).
d. Can you come up with a vector
\[
\begin{bmatrix}
a \\
b \\
c
\end{bmatrix}
\]
that is not in \( V \)? Is this okay?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa00d0151-b5df-4ee3-a0d6-fdfba7d1c4f7%2F81a64680-d1e0-498c-8a10-d0c450b1faf5%2Fni98u9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Vector Space Problem
1. Let
\[
V = \left\{ \begin{bmatrix} 1 + t \\ 2 - t \\ 3 + 2t \end{bmatrix} \mid t \in \mathbb{R} \right\}
\]
Define
\[
\begin{bmatrix}
1 + t_1 \\
2 - t_1 \\
3 + 2t_1
\end{bmatrix}
\oplus
\begin{bmatrix}
1 + t_2 \\
2 - t_2 \\
3 + 2t_2
\end{bmatrix}
=
\begin{bmatrix}
1 + (t_1 + t_2) \\
2 - (t_1 + t_2) \\
3 + 2(t_1 + t_2)
\end{bmatrix}
\]
and
\[
c \odot
\begin{bmatrix}
1 + t \\
2 - t \\
3 + 2t
\end{bmatrix}
=
\begin{bmatrix}
1 + ct \\
2 - ct \\
3 + 2ct
\end{bmatrix}
\]
#### Tasks:
a. Find the additive identity and additive inverse.
b. Show that \( V \) is a vector space (does it comply with ALL ten properties of a vector space?).
c. Verify that \( 0 \cdot \mathbf{v} = \mathbf{0} \) for all \(\mathbf{v}\).
d. Can you come up with a vector
\[
\begin{bmatrix}
a \\
b \\
c
\end{bmatrix}
\]
that is not in \( V \)? Is this okay?
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