# Prove that V is the internal direct sum of two subspaces

Austin Chang
Let V be a vector space. If U 1 and U2 are subspaces of V s.t. U1+U2 = V and U1 and U1∩U2 = {0V}, then we say that V is the internal direct sum of U1 and U2. In this case we write V = U1⊕U2. Show that V is internal direct sum of U1 and U2if and only if every vector in V may be written uniquely in the form v1+v2 with v1∈U1 and v2∈ U2.

What does it mean to be unique? Does it matter if it is unique?

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What does it mean to be unique?
It means that for any vector ##u\in V##, if ##w_1+w_2=u=v_1+v_2##, for ##w_1,v_1\in U_1## and ##w_2,v_2\in U_2##, then ##v_1=w_1## and ##v_2=w_2##.

Yes, the uniqueness matters. If the two subspaces overlap, there is more than one way of writing a vector in ##V## as a sum of two vectors one from each subspace. And vice versa, if the representation as a sum is not unique, the two subspaces must overlap.

Austin Chang
It means that for any vector ##u\in V##, if ##w_1+w_2=u=v_1+v_2##, for ##w_1,v_1\in U_1## and ##w_2,v_2\in U_2##, then ##v_1=w_1## and ##v_2=w_2##.

Yes, the uniqueness matters. If the two subspaces overlap, there is more than one way of writing a vector in ##V## as a sum of two vectors one from each subspace. And vice versa, if the representation as a sum is not unique, the two subspaces must overlap.
So for your example if it was not unique U2 = U1 and you can write w1 in U2 and w2 in U1? therefore w2+w1 is not unique anymore because the same things came from different subspaces?

Let V be a vector space. If U 1 and U2 are subspaces of V s.t. U1+U2 = V and U1 and U1∩U2 = {0V}, then we say that V is the internal direct sum of U1 and U2. In this case we write V = U1⊕U2. Show that V is internal direct sum of U1 and U2if and only if every vector in V may be written uniquely in the form v1+v2 with v1∈U1 and v2∈ U2.

What does it mean to be unique? Does it matter if it is unique?

Perhaps an example of non-uniqueness will make this clear. You can represent 12 as the product of two factors ##12 = x_1 x_2## but the two factors are not unique since they and be chosen to have various different values - e.g. (12)(1), (4)(3), (2)(6) etc.

As whether uniqueness matters - that's a subjective question. Uniqueness is often a convenient property to have. For example, the rigorous approach to defining "a" zero z_1 in arithmetic would be to define it by the property that for all numbers x, z1 + x = x. It is convenient that there is a unique number, denoted by "0", that has this property. If there were several unequal zeroes, arithmetic would be more complicated.

Austin Chang
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So for your example if it was not unique U2 = U1
No, that would be going too far. U2 and U1 will overlap nontrivially, but that doesn't mean they are identical. All it means that their intersection is a vector space of dimension at least 1.

Austin Chang
Perhaps an example of non-uniqueness will make this clear. You can represent 12 as the product of two factors ##12 = x_1 x_2## but the two factors are not unique since they and be chosen to have various different values - e.g. (12)(1), (4)(3), (2)(6) etc.

As whether uniqueness matters - that's a subjective question. Uniqueness is often a convenient property to have. For example, the rigorous approach to defining "a" zero z_1 in arithmetic would be to define it by the property that for all numbers x, z1 + x = x. It is convenient that there is a unique number, denoted by "0", that has this property. If there were several unequal zeroes, arithmetic would be more complicated.
Thanks! That was a good example.

Austin Chang
No, that would be going too far. U2 and U1 will overlap nontrivially, but that doesn't mean they are identical. All it means that their intersection is a vector space of dimension at least 1.
What do you mean by at least a vector space of dimension 1? I don't think I've gotten that far to understand it completely. Could you elaborate?