Question

Let V = R 3 , A = {(x, y,z)|y = 0} and B = {(x, y,z)|x = y = z}. Check whether R 3 = A⊕B ?

Answer 1

Kindly refer to the Explanation.

Explanation:

Recall that, `Ao+B={alpha+beta | alpha in A, beta inB}.`

Clearly, `Ao+B sub R^3`.

So, in order to show that, `R^3=Ao+B`, it suffices to show that,

`R^3 sub Ao+B.............(star)`.

Let. `xi=(x',y',z') in R^3, xi" arbitrary"`.

Consider, `alpha=(x'-y',0,z'-y') and beta=(y',y',y')`.

`"Evidently, "alpha in A, and beta in B," and we have, "`

`alpha+beta=(x'-y',0,z'-y')+(y',y',y')=(x', y',z')=xi`.

`"Thus, "xi in R^3 rArr xi in Ao+B`.

This proves `(star)`.

Hence, the verification that, `R^3=Ao+B`.