Question

4. (10 points) Determine if the subsets $W$ of $\mathbb{R}^3$ defined below are subspaces of $\mathbb{R}^3$. If so, give a proof; if not, explain why. (a) $W = \left\{ \begin{bmatrix} t+s \\ 5t \\ t+3s \end{bmatrix} \in \mathbb{R}^3 \right\}$. (b) $W = \left\{ \begin{bmatrix} a \\ b \\ c \end{bmatrix} \in \mathbb{R}^3 : b+c \ge 0 \right\}$.

Name: 4 10 points determine if the subsets w of mathbbr3 defined below are subspaces of mathbbr3 if so give a proof if not explain why a w left beginbmatrix ts 5t t3s endbmatrix in mathbbr3 right 44775
Uploaded: 2024-05-09T11:50:31-08:00
Duration: 3 min 35 s
Channel: Adi S
Description: 4 10 points determine if the subsets w of mathbbr3 defined below are subspaces of mathbbr3 if so give a proof if not explain why a w left beginbmatrix ts 5t t3s endbmatrix in mathbbr3 right 44775

          4. (10 points)
Determine if the subsets $W$ of $\mathbb{R}^3$ defined below are subspaces of $\mathbb{R}^3$.
If so, give a proof; if not, explain why.
(a) $W = \left\{ \begin{bmatrix} t+s \\ 5t \\ t+3s \end{bmatrix} \in \mathbb{R}^3 \right\}$.
(b) $W = \left\{ \begin{bmatrix} a \\ b \\ c \end{bmatrix} \in \mathbb{R}^3 : b+c \ge 0 \right\}$.

4 10 points determine if the subsets w of mathbbr3 defined below are subspaces of mathbbr3 if so give a proof if not explain why a w left beginbmatrix ts 5t t3s endbmatrix in mathbbr3 right 44775

Added by Laura G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

05/09/2024

Step 1

Let $c_1, c_2 \in \mathbb{R}$. Then $c_1\mathbf{w}_1 + c_2\mathbf{w}_2 = c_1\begin{bmatrix} t_1+s_1 \\ 5t_1 \\ t_1+3s_1 \end{bmatrix} + c_2\begin{bmatrix} t_2+s_2 \\ 5t_2 \\ t_2+3s_2 \end{bmatrix} = \begin{bmatrix} c_1(t_1+s_1) + c_2(t_2+s_2) \\ 5(c_1t_1 + c_2t_2) Show more…

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4. (10 points) Determine if the subsets $W$ of $\mathbb{R}^3$ defined below are subspaces of $\mathbb{R}^3$. If so, give a proof; if not, explain why. (a) $W = \left\{ \begin{bmatrix} t+s \\ 5t \\ t+3s \end{bmatrix} \in \mathbb{R}^3 \right\}$. (b) $W = \left\{ \begin{bmatrix} a \\ b \\ c \end{bmatrix} \in \mathbb{R}^3 : b+c \ge 0 \right\}$.