Question

Let V = R3
, A = {(x, y,z)|y = 0} and B = {(x, y,z)|x = y = z}. Check whether
R3 = A⊕B

Answer 1

Answer on Question #82544 – Math – Linear Algebra

Question

Let $V = \mathbb{R}^3$, $A = \{(x,y,z) | y = 0\}$ and $B = \{(x,y,z) | x = y = z\}$.

Check whether $\mathbb{R}^3 = A \oplus B$.

Solution

$C = A \oplus B$ if and only if $C = A + B$ and $A \cap B = 0$.

1) $(x, y, z) \in A \cap B$ if and only if $(x, y, z) \in A$ and $(x, y, z) \in B$, so

If $(x,y,z)\in B$ then $x = y = z$ so $(x,y,z) = (x,x,x)$ and if $(x,x,x)\in A$ then $x = 0$ so $(x,y,z) = (0,0,0)$, so $A\cap B = 0$

2) $(x,y,z) = (x - y + y, y, z - y + y) = (x - y, 0, z - y) + (y, y, y)$

$(x - y, 0, z - y) \in A$ and $(y, y, y) \in B$ so $\mathbb{R}^3 = A + B$

So, both conditions are true and $\mathbb{R}^3 = A \oplus B$.

**Answer**: $\mathbb{R}^3 = A \oplus B$ holds true.

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