Question

Let F be a field. If A = {(x, y, 0): x, y ∈ F}, B = {(0, y, z): y, z ∈F} be subspaces of F³(F), find the dimension of the subspace A+B

Answer 1

Given:

F be a field

A={(x,y,0):x,y$\in F$}

B={(0,y,z):y,z$\in F$}

are sub-spaces of F3(F).

To find:

Dimension of the subspace of A+B

Solution:

Let an element a=(x,y,0) $\in A$ and b=(0,y,z)$\in B$

$a+b=(x,y,0)+(0,y,z)$

$a+b=(x,y+y,z)=(x,2y,z)$

Dimension of subspace  A+B=3