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Hi, everybody!

I'd like to ask you about the direct sum of subspaces...

I refer to two linear algebra books; 1)Friedberg's book, 2)Hoffman's book.

First of all, I write two definitions of direct sum of subspaces...

in the book 1),

Def.1). Let W[itex]_{1}[/itex],...,W[itex]_{k}[/itex] be subspaces of a vector space V. We call V the direct sum of the subspaces W[itex]_{1}[/itex],...,W[itex]_{k}[/itex]

and write V=W[itex]_{1}[/itex][itex]\oplus[/itex]...[itex]\oplus[/itex]W[itex]_{k}[/itex], if

V = [itex]\sum[/itex][itex]^{k}_{i=1}[/itex]W[itex]_{i}[/itex] and

W[itex]_{j}[/itex][itex]\cap[/itex][itex]\sum[/itex][itex]_{i\neq j}[/itex]W[itex]_{i}[/itex]={0} for each j(1[itex]\leq[/itex]j[itex]\leq[/itex]k);I intepret this condition by logical form as follows;[itex]\forall[/itex]j(1[itex]\leq[/itex]j[itex]\leq[/itex]k and j is an integer [itex]\rightarrow[/itex]W[itex]_{j}[/itex][itex]\cap[/itex][itex]\sum[/itex][itex]_{i\neqj}[/itex]W[itex]_{i}[/itex]={0}).

In the book 2),

Def.2). Let W[itex]_{1}[/itex],...,W[itex]_{k}[/itex] be subspaces of a finite-dimensional vector space V. We call V the direct sum of the subspaces W[itex]_{1}[/itex],...,W[itex]_{k}[/itex]

and write V=W[itex]_{1}[/itex][itex]\oplus[/itex]...[itex]\oplus[/itex]W[itex]_{k}[/itex],

if V = [itex]\sum[/itex][itex]^{k}_{i=1}[/itex]W[itex]_{i}[/itex] and the subspaces have

the property such that for each j, 2[itex]\leq[/itex]j[itex]\leq[/itex]k, we have W[itex]_{j}[/itex][itex]\cap[/itex](W[itex]_{1}[/itex]+...+W[itex]_{j-1}[/itex])={0};I interpret this condition by logical form as follows; [itex]\forall[/itex]j(2[itex]\leq[/itex]j[itex]\leq[/itex]k and j is an integer [itex]\rightarrow[/itex]W[itex]_{j}[/itex][itex]\cap[/itex](W[itex]_{1}[/itex]+...+W[itex]_{j-1}[/itex])={0}).

Now, as you can see, in Def.1), since [itex]\sum[/itex][itex]_{i\neqj}[/itex]W[itex]_{i}[/itex] does not exist when k=1, I cannot determine whether direct sum of subspaces is defined or not when k=1.

On the other hand, in the second definition; that is, Def.2), from the logical form, I can see that it is vacuously true when k=1. So, in this case, I can say V is a direct sum of its subspace V=W[itex]_{1}[/itex]. I think this means that direct sum of subspaces can be defined even when k=1.

How do you think about Def.1)? Do you think that there're some mistakes in the reasoning I've suggested above?

To sum up, I wanna ask you if 1. direct sum can be defined even when k=1

2. if so, where do I make such mistakes??

I hope you help me solve this problem...

Thank you for reading my long questions...

Have a nice day!!!

(If you want to have more specific information of definitions of them, refer to p. 275 in 1) and p. 219~220 in 2).)

2. Relevant equations

Red lines above indicate

i [itex]\neq[/itex] j

3. The attempt at a solution

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# The direct sum of subspaces

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