Is it true that subspaces satisfy the distributive law M N1

Is it true that subspaces satisfy the distributive law M (N_1 + N_2) = M N_1 + M N_2? If not, give a counter example.

Solution

Consider the vector space R². Let N₁ and N₂ be the subspaces spanned by the standard vectors ₁and ₂ respectively. Then N₁ + N₂ = R².

Next, let M denote the subspace spanned by ₁ + ₂.

Then M ( N₁ + N₂ ) = M R² = M

But M N₁ = M N₂ = { 0 } , the zero subspace of R²,

so that ( M N₁ ) + ( M N₂ ) = { 0 }.

That is M ( N₁ + N₂ ) ( M N₁ ) + ( M N₂ ).

It only holds in the trivial cases dim V 1.

Ex :-

If dimV 2, take S, T, U to be three different subspaces such that S U + T ,

that is, S lies in the subspace generated by U and T.

Then S ( U + T ) = S

but ( S U ) + ( S T ) = 0 + 0 = 0 .

Hence S ( U + T )    ( S U ) + ( S T ).