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I understand a couple people have posted about this before, but different sources are giving me different answers on what exactly a direct sum is with regards to vector spaces. I've seen both of these definitions for $V \oplus W$:

$$V \oplus W = \{ (v,w) \mid v \in V, w \in W \}$$

$$V \oplus W = \{ v + w \mid v\in V, w\in W, v+w \space \text{distinct} \}$$

These definitions are all given that addition and scalar multiplication done component-wise. In the second definition, different sources note that $v + w$ being distinct is only for the direct sum. Is $V \oplus W$ a $2$-tuplet, or a set of vectors like $V$ and $W$ are?

In addition, if $V$ has dimension $x$, and $W$ has dimension $y$, what is the dimension of $V \oplus W$, and why?

I'm actually pretty certain the definitions I gave above are imprecise and somewhat incorrect, but if someone could help me with this, that'd be great. Thanks.

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    $\begingroup$ If $v,w$ are vectors, what does "$v+w$ distinct" mean? $\endgroup$
    – coffeemath
    Dec 6, 2020 at 1:07
  • $\begingroup$ That all entries in the set are distinct @coffeemath $\endgroup$
    – Luna145
    Dec 6, 2020 at 1:08
  • $\begingroup$ Luna--- That would make it hard to deal with a direct sum made from two subspaces of one vector space. $\endgroup$
    – coffeemath
    Dec 6, 2020 at 1:12
  • $\begingroup$ Then I suppose that is the wrong definition. Are you able to give some more information? $\endgroup$
    – Luna145
    Dec 6, 2020 at 1:13
  • $\begingroup$ I included a couple of examples as an addition to the answer, which hopefully make it clear. $\endgroup$ Dec 6, 2020 at 2:07

2 Answers 2

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The first definition is of an external direct sum, whereas the second definition is of an internal direct sum. See here, for example. These definitions are isomorphic.

The external direct sum does result in tuples. The dimension in this case sum since the tuples are the result of the Cartesian product of the basis vectors.

External direct sums builds up new vector spaces. For example, the vector space of polynomials of the form $a_0+a_1x+a_2x^2$ has basis $V=\left\{1,x,x^2\right\}$ can be direct summed to the vector space of polynomials of the form $b_3x^3+b_4x^4+b_5x^5$ with basis $W=\left\{x^3,x^4,x^5 \right\}$ to yield the vector space $V \oplus W$ of polynomials of the form $c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5$ of dimension $\small \text{dim}(V) + \text{dim}(W)=6.$

The vector spaces added in the external direct sum $V$ and $W$ do not belong to the sum. For instance the associated vector space to $\mathbb R^2$ can be direct summed to $\mathbb R^3$ to yield $\mathbb R^5.$ The basis vectors of this external direct sum are represented in a block matrix (assuming the standard basis for each one) - the basis vectors of $\mathbb R^2$ are "padded" with three added zeros at the bottom, while the basis vectors of $\mathbb R^3$ need two added zeros at the top:

$$\small\begin{bmatrix}1&0&&&\\0&1&&&\\&&1&0&0\\&&0&1&0\\&&0&0&1 \end{bmatrix}$$

The dimension stays unchanged in the interior direct sum if $V$ and $W$ are subspaces without intersection other than the zero vector.

Example: In this case $V$ may be the subspace of $\mathbb R^3$ given by vectors of the form $\small\begin{bmatrix}a\\b\\0 \end{bmatrix}$ and $W$ be the subgroup of vectors of the form $\small\begin{bmatrix}0\\0\\c \end{bmatrix}.$ The internal direct sum will span the ambient space, and be formed by vectors of $\small\begin{bmatrix}a\\b\\0 \end{bmatrix} + \small\begin{bmatrix}0\\0\\c \end{bmatrix}=\small\begin{bmatrix}a\\b\\c \end{bmatrix}$ where $a,b,c\in \mathbb R.$

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Your first definition is the correct definition (along with a suitable definition of vector addition and scaling). It works for any two vector spaces $V$ and $W$.

On the other hand, if $V$ and $W$ are both subspaces of the same, larger vector space $U$, then $v+w$ for $v\in V, w\in W$ makes sense to talk about. With this, we can construct $V+W$ as the subspace $$ V+W=\{v+w\mid v\in V, w\in W\}\subseteq U $$ Then one can ask: When is $V\oplus W$ isomorphic to (i.e. "essentially the same as") $V+W$? And the answer is when $V\cap W=\{0\}$. When this is true, $V+W$ is also called "the direct sum", because it's a bit pedantic to insist on maintaining the distinction (at least once they have been shown to be isomorphic), and it's often more hassle than its worth to keep using the ordered pairs when we already have the elements of $U$ to work with.

Direct sums add together dimensions. it's not difficult to see that if $\{v_1, v_2, \ldots, v_x\}$ is a basis for $V$ and $\{w_1,\ldots,w_y\}$ is a basis for $W$, then $$ \{(v_1,0),(v_2,0),\ldots,(v_x,0),(0,w_1),(0,w_2),\ldots,(0,w_y)\} $$ is a basis for $V\oplus W$.

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