Answer to Question #163675 in Linear Algebra for Nikhil Rawat

Question

Let V= R^3

W={(x1, x2, x3)| x1-x2 =x3}. Show that W is a

subspace of V. Further,find a basis for W and hence,find the dimension of W.

Answer 1

Let V= R^3

$W={(x_1, x_2, x_3)| x_1-x_2 =x_3}$ . Show that W is a

subspace of V.

Further,find a basis for W and hence,find the dimension of W.

To show that W is a subspace of V we need to show the following:

let $r= (r_1, r_2, r_3)$ , $q=(q_1, q_2,q_3) \isin W$ and $\alpha\isin$ field

$\implies q_1-q_2=q_3$ and $r_1- r_2= r_3$

1. W is not empty
2. for r,q $\isin$ W , r+q $\in$ W
3. $\alpha r\isin W$

if $(0, 0,0)\isin$ W then $0-0=0$

therefore, W is not empty

$r+q=(r_1, r_2, r_3) +(q_1, q_2,q_3)={ (r_1+q_1), (r_2+q_2),(r_3+q_3)}$

since $(r_1+q_1)-(r_2+q_2)=(r_1-r_2) +(q_1-q_2)= r_3+ q_3$

$\implies r+q\isin W$

$\alpha r=\alpha (r_1, r_2, r_3)=r= (\alpha r_1, \alpha r_2,\alpha r_3)$

then $\alpha r_1-\alpha r_2=\alpha(r_1-r_2)=\alpha r_3$

$\implies \alpha r\isin W$

therefore W is a subspace of V

$r=\begin{pmatrix} r_1 \\ r_2 \\r_3 \end{pmatrix} =\begin{pmatrix} r_1 \\ r_2 \\r_1-r_2 \end{pmatrix} = r_1\begin{pmatrix} 1 \\ 0 \\1 \end{pmatrix}+r_2\begin{pmatrix} 0 \\ 0 \\-1 \end{pmatrix}$

$\therefore , \begin{bmatrix} \begin{pmatrix} 1 \\ 0 \\1 \end{pmatrix} & \begin{pmatrix} 0 \\ 0 \\-1 \end{pmatrix} \end{bmatrix}$ is a Basis of W

The Dimension of W is 2