Answer to Question #222298 in Linear Algebra for MAXINE

Question

Determine whether the subset A={(x,y,z):y=x+1 and z=y+1} is a subspace of R³

Answer 1

Solution;

By definition;

W is a subspace if;

1.A $\neq$ $\empty$

2. A is closed under vector addition.

Test for both conditions;

Clear A is not empty .

Check for vector addition;

If you take arbitrary values ,for example, u= (1,2,3) to satisfy A, since;

y=2=x+1=1+1

z=3=y+1=2+1

Also ,we take another set of arbitrary values which also satisfy A;

v=(0,1,2) $\in$ A;

If we add u and v;

(u+v)=(1,3,5)

From A,

y=x+1 and z=y+1

By using (u+v)

y=1+1=2

2$\neq$ 3

Also ;

z=y+1=2+1=3

3$\neq$ 5

Hence A is not a subspace in R³