Question

Problem 1. Determine whether the sets $ U $ are $ V $ are subspace of $ \mathbb{R}^{4} $ defined by \[ \begin{array}{l} U=\left\{(x, y, z, \theta) \in \mathbb{R}^{4}: x-y-z\right\} \text { and } \\ V^{*}=\left\{(x, y, z, v) \in \mathbb{R}^{4}: x=2 z-1 \text { and } y=v\right\} . \end{array} \]

          Problem 1. Determine whether the sets \( U \) are \( V \) are subspace of \( \mathbb{R}^{4} \) defined by
\[
\begin{array}{l}
U=\left\{(x, y, z, \theta) \in \mathbb{R}^{4}: x-y-z\right\} \text { and } \\
V^{*}=\left\{(x, y, z, v) \in \mathbb{R}^{4}: x=2 z-1 \text { and } y=v\right\} .
\end{array}
\]

$Problem 1. Determine whether the sets U are V are subspace of ℝ^4 defined by U={(x, y, z, θ) ∈ℝ^4: x-y-z} and V^*={(x, y, z, v) ∈ℝ^4: x=2 z-1 and y=v} .$

Added by Tammy J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Tim Thornhill

Step 1

- **Condition 1: Non-emptiness (contains the zero vector).** The zero vector in $ \mathbb{R}^4 $ is $(0, 0, 0, 0)$. Check if it satisfies the condition $ x = y - z $. Substitute: $ 0 = 0 - 0 $, which is true. Therefore, $ U $ contains the zero Show more…

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Problem 1. Determine whether the sets $ U $ are $ V $ are subspace of $ \mathbb{R}^{4} $ defined by \[ \begin{array}{l} U=\left\{(x, y, z, \theta) \in \mathbb{R}^{4}: x-y-z\right\} \text { and } \\ V^{*}=\left\{(x, y, z, v) \in \mathbb{R}^{4}: x=2 z-1 \text { and } y=v\right\} . \end{array} \]