Properties of Finite Sums of Vectors

Let $K$ be a field and let $V$ be a vector space over $K$ with zero vector $0_{V}$. Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ as in that definition, ordered by the relation $\le$ of that definition, let $n\in\mathbb{N}$, and let $[n]$ be the initial segment determined by $n$, that is, the set of natural numbers $k$ with $1\le k\le n$. Let $u:[n]\to V$ and $v:[n]\to V$ be maps with values $u_{k}$ and $v_{k}$, let $c:[n]\to K$ be a map with values $c_{k}$, and let $\lambda\in K$.

Sums of vectors are the finite sums in a vector space, and sums of scalars are the finite sums in a field. Then the following hold.

1. (Restriction and recursion) If $j\in[n]$ and $v'$ denotes the restriction of $v$ to $[j]$, then

$$\sum_{k=1}^{i}v'_{k}=\sum_{k=1}^{i}v_{k}\qquad\text{for every }i\in[j].$$

Moreover

$$\sum_{k=1}^{1}v_{k}=v_{1},\qquad \sum_{k=1}^{S(m)}v_{k}=\Bigl(\sum_{k=1}^{m}v_{k}\Bigr)+v_{S(m)}\quad\text{whenever }S(m)\in[n].$$

2. (Additivity)

$$\sum_{k=1}^{n}(u_{k}+v_{k})=\sum_{k=1}^{n}u_{k}+\sum_{k=1}^{n}v_{k}.$$

3. (Homogeneity)

$$\sum_{k=1}^{n}(\lambda v_{k})=\lambda\sum_{k=1}^{n}v_{k}.$$

4. (Linear maps) Let $W$ be a vector space over $K$ and let $T:V\to W$ be a linear map. Then

$$T\Bigl(\sum_{k=1}^{n}v_{k}\Bigr)=\sum_{k=1}^{n}T(v_{k}),$$

the sum on the right being the finite sum in $W$.

5. (Inner products against a finite sum) Suppose $K$ is the field of complex numbers and $V$ together with $\langle\cdot,\cdot\rangle$ is a complex inner product space, and let $w\in V$. Then

$$\Bigl\langle w,\sum_{k=1}^{n}v_{k}\Bigr\rangle=\sum_{k=1}^{n}\langle w,v_{k}\rangle,\qquad \Bigl\langle \sum_{k=1}^{n}v_{k},w\Bigr\rangle=\sum_{k=1}^{n}\langle v_{k},w\rangle,$$

the sums on the right being finite sums in the field of complex numbers.

6. (Inner products against a linear combination) Under the hypotheses of claim 5, with the complex conjugate,

$$\Bigl\langle w,\sum_{k=1}^{n}c_{k}v_{k}\Bigr\rangle=\sum_{k=1}^{n}c_{k}\langle w,v_{k}\rangle,\qquad \Bigl\langle \sum_{k=1}^{n}c_{k}v_{k},w\Bigr\rangle=\sum_{k=1}^{n}\overline{c_{k}}\,\langle v_{k},w\rangle .$$

7. (A single possibly nonzero summand) Let $i\in[n]$ and suppose that $v_{k}=0_{V}$ for every $k\in[n]$ with $k\ne i$. Then

$$\sum_{k=1}^{n}v_{k}=v_{i}.$$

In particular, if $v_{k}=0_{V}$ for every $k\in[n]$, then $\sum_{k=1}^{n}v_{k}=0_{V}$.