Question

V04 (version 967): Consider the following two sets of Euclidean vectors: -- {0) --***-o} - {:}------ Explain why one of these sets is a subspace of R3, and why the other is not.

Answer 1

$U= \left \{ \begin{bmatrix} x\\y \\ z \end{bmatrix} \ : \ -2x-y+4z=0 \right \}$ and   $W= \left \{ \begin{bmatrix} x\\y \\ z \end{bmatrix} \ : \ -2|z|=4x-y \right \}$

Which one of these set is subspace of $\mathbb{R}^3$ ?

Solution :

Consider the set   $U= \left \{ \begin{bmatrix} x\\y \\ z \end{bmatrix} \ : \ -2x-y+4z=0 \right \}$  

By the definition of set    , the elements in    are of the form $-2x-y+4z=0$ .

Now $-2x-y+4z=0$ is the equation of plane passing through origin .

And any plane passing through origin is a subspace of $\mathbb{R}^3$ .

Therefore elements of forms a subspace of  $\mathbb{R}^3$ .

Hence  

$U= \left \{ \begin{bmatrix} x\\y \\ z \end{bmatrix} \ : \ -2x-y+4z=0 \right \}$   is subspace of $\mathbb{R}^3$ .

Now Consider the set $W= \left \{ \begin{bmatrix} x\\y \\ z \end{bmatrix} \ : \ -2|z|=4x-y \right \}$ .

Consider the elements  $\begin{bmatrix} -1\\ 0 \\ -2 \end{bmatrix} \ \ and \ \ \begin{bmatrix} -1\\ 0 \\ 2 \end{bmatrix}$

WE can see that  

$\begin{bmatrix} -1\\ 0 \\ -2 \end{bmatrix} \in W \ \ and \ \ \begin{bmatrix} -1\\ 0 \\ 2 \end{bmatrix} \in W$

Now consider

$\begin{bmatrix} -1\\ 0 \\ -2 \end{bmatrix} \ \ +\ \ \begin{bmatrix} -1\\ 0 \\ 2 \end{bmatrix} \ = \ \begin{bmatrix} -2\\ 0 \\ 0 \end{bmatrix}$

But  $\begin{bmatrix} -2\\ 0 \\ 0 \end{bmatrix} \notin W$  $...............(\ because \ \ -2|0|\neq4(-2)-0 \ \ )$

Therefore we get  

$\begin{bmatrix} -1\\ 0 \\ -2 \end{bmatrix} \in W \ \ and \ \ \begin{bmatrix} -1\\ 0 \\ 2 \end{bmatrix} \in W$ But   $\begin{bmatrix} -1\\ 0 \\ -2 \end{bmatrix} \ \ +\ \ \begin{bmatrix} -1\\ 0 \\ 2 \end{bmatrix} \ \ \ \notin W$

Therefore is not closed under addition .

Therefore    is not a subspace of   $\mathbb{R}^3$ .

$W= \left \{ \begin{bmatrix} x\\y \\ z \end{bmatrix} \ : \ -2|z|=4x-y \right \}$    is not a subspace of $\mathbb{R}^3$ .

Hence  

$U= \left \{ \begin{bmatrix} x\\y \\ z \end{bmatrix} \ : \ -2x-y+4z=0 \right \}$   is subspace of $\mathbb{R}^3$ .

AND

$W= \left \{ \begin{bmatrix} x\\y \\ z \end{bmatrix} \ : \ -2|z|=4x-y \right \}$    is not a subspace of $\mathbb{R}^3$ .