# The definition of direct sums and subspaces?

From what I have read the definition of the direct sum of the vector spaces $V_1$, $V_2$ is the set $V_1\times V_2$ with the operations of addition and scalar multiplication defined as follows (Knapp, 2006): $(u_1,u_2)+(v_1,v_2)=(u_1+v_1,u_2+v_2)$ $c(v_1,v_2)=(cv_1,cv_2)$ But then today I came across the fact that if $V$ has linear subspaces $V_1$, $V_2$ and every $v\in V$ can be written uniquly as: $$v=v_1+v_2$$ for $v_i \in V_i$ then $V=V_1\oplus V_2$. I cannot see how this makes sense given the definition above. Is the above definition wrong? If not how can $V$ have the form $(v_1,v_2)$ when $v_1,v_2 \in V$?

• The direct sum should be called "direct product".
– Xam
Aug 2, 2017 at 14:13
• @Xam is there a difference? Aug 2, 2017 at 14:16
• We can say that $V$ is isomorphic to the direct sum of $V_1$ and $V_2$. Aug 2, 2017 at 15:42
• See the direct sum definition here : web.iitd.ac.in/~ritumoni/lecture%2011%20&%2012.pdf Jun 29, 2023 at 5:13

The sum of two subspaces of a vector space $V_1,V_2\subseteq V$ (denoted as $V_1+V_2$ is defined as

$$V_1 + V_2 = \{v_1 + v_2| v_1\in V_1\land v_2\in V_2\}$$

and it is a vector subspace of $V$.

The direct sum of two vector spaces $V_1, V_2$, (denoted as $V_1\oplus V_2$) is a vector space $V_1\times V_2$ with operations defined as you wrote.

The connection between a direct and ordinary sum is that if $V_1, V_2\subset V$ and $V_1$ and $V_2$ are linearly independent, then $V_1+V_2$ is isomorphic to $V_1\oplus V_2$.

• Is this definition of the direct sum the most general? i.e. does it work for both finite and infinite vector spaces? Aug 2, 2017 at 14:03
• @Quantumspaghettification Yes, there is no difference between the two in this regard. Aug 2, 2017 at 14:03

The $+$ is defined on pairs of subspaces of a given vector space and $\times$ on pairs of vector spaces (not considered as subspaces of any sorts) and we write $V_1 \oplus V_2$ in place of $V_1 + V_2$, if $V_1 \cap V_2 = \{0\}$.

These two things are "the same" in the sense, that if $V_1,V_2$ are subvectorspaces of $V$ with $V_1 \cap V_2 = \{0\}$, then $V_1 \oplus V_2$ is isomorphic to $V_1 \times V_2$ which is not hard to prove.

• Do you mean then $V_1\oplus V_2$ is isomorphic to $V_1 +V_2$ instead of $V_1 \times V_2$? Aug 2, 2017 at 14:07
• @Quantumspaghettification They are the same. The notation $V_1 \oplus V_2$ in place of $V_1 + V_2$, if $V_1 \cap V_2 = \{0\}$, is just very popular (I guess to be brief? It's the same idea as disjoint unions of subsets, c.f. math.stackexchange.com/questions/1631396/…). EDIT: sorry, I misread your comment. I mean: $V_1 \oplus V_2 = V_1 + V_2 \cong V_1 \times V_2$ ($\cong$ meaning "is isomorphic to") Aug 2, 2017 at 14:11

I have just found this pdf: http://www.math.ncku.edu.tw/~fjmliou/advcal/sumvspace.pdf which summarizes direct sums, and explains what is going on here. I will here outline what it says. Thanks also to the other answers which helped clarify things.

This is to do with the difference between an internal and external direct product:

# The External Direct Sum

Let $$V$$ and $$W$$ be two vector spaces over $$F$$ then we the external direct product $$V\oplus_e W$$ is defined as I have given it in the question. i.e. the vector space with underlying set $$V\times W$$ and operators defined by: $(u_1,u_2)+(v_1,v_2)=(u_1+v_1,u_2+v_2)$ $c(v_1,v_2)=(cv_1,cv_2)$

# The Internal Direct Sum

For $$V_1$$ and $$V_2$$ subspaces of $$V$$ which satisfy:

• for each $$v\in V$$ there exists $$v_1 \in V_1$$ and $$v_2 \in V_2$$ such that $$v=v_1+v_2$$.
• $$V_1 \cap V_2\{0\}$$

in this case we write $$V=V_1 \oplus_i V_2$$ and $$V$$ is the inner direct product of $$V_1$$ and $$V_2$$.

There is a linear isomorphism such that $$V_1\oplus_e V_2$$ is isomorphic to $$V_1 \oplus_i V_2$$