Let $𝑉$ be a vector space over a field $𝐹$, and let $\mathrm{𝐯}\in 𝑉$ be an arbitrary vector. Call the field additive identity $0_{𝐹}\in 𝐹$. Then:

Let $𝑉$ be a vector space over a field $𝐹$, and let $𝑎\in 𝐹$ be an arbitrary scalar. Call the vector additive identity ${\mathrm{𝟎}}_{𝑉}\in 𝑉$. Then:

Let $𝑉$ be a vector space over a field $𝐹$, and let $\mathrm{𝐯}\in 𝑉$ be an arbitrary vector. Call the field additive identity $0_{𝐹}\in 𝐹$ and the field multiplicative identity $1\in 𝐹$. By the definition of the additive inverse, there exists a scalar $-1\in 𝐹$ such that $1+\left(-1\right)=0_{𝐹}$. Then:

Let $𝐹$ be a field, and let ${𝐹}^{𝑛}$ be the set as defined in Definition 7. Define vector addition and scalar multiplication on ${𝐹}^{𝑛}$ as in Definition 8. We will verify that ${𝐹}^{𝑛}$ satisfies all vector space axioms under these operations.

Let $𝑎,𝑏\in 𝐹$ be arbitrary scalars, and let $\mathrm{𝐮}=\left({𝑢}_1,{𝑢}_2,\dots ,{𝑢}_{𝑛}\right),\mathrm{𝐯}=\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right),\mathrm{𝐰}=\left({𝑤}_1,{𝑤}_2,\dots ,{𝑤}_{𝑛}\right)\in {𝐹}^{𝑛}$ be arbitrary vectors. Then:

1. Associativity of vector addition.

   $$\begin{array}{cc}\left(\mathrm{𝐮}+\mathrm{𝐯}\right)+\mathrm{𝐰}& =\left(\left({𝑢}_1,{𝑢}_2,\dots ,{𝑢}_{𝑛}\right)+\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\right)+\left({𝑤}_1,{𝑤}_2,\dots ,{𝑤}_{𝑛}\right)\\ & =\left(\left({𝑢}_1+{𝑣}_1\right)+{𝑤}_1,\left({𝑢}_2+{𝑣}_2\right)+{𝑤}_2,\dots ,\left({𝑢}_{𝑛}+{𝑣}_{𝑛}\right)+{𝑤}_{𝑛}\right)\\ & =\left({𝑢}_1+\left({𝑣}_1+{𝑤}_1\right),{𝑢}_2+\left({𝑣}_2+{𝑤}_2\right),\dots ,{𝑢}_{𝑛}+\left({𝑣}_{𝑛}+{𝑤}_{𝑛}\right)\right)\\ & =\left({𝑢}_1,{𝑢}_2,\dots ,{𝑢}_{𝑛}\right)+\left({𝑣}_1+{𝑤}_1,{𝑣}_2+{𝑤}_2,\dots ,{𝑣}_{𝑛}+{𝑤}_{𝑛}\right)\\ & =\mathrm{𝐮}+\left(\mathrm{𝐯}+\mathrm{𝐰}\right).\end{array}$$

2. Commutativity of vector addition.

   $$\begin{array}{cc}\mathrm{𝐮}+\mathrm{𝐯}& =\left({𝑢}_1,{𝑢}_2,\dots ,{𝑢}_{𝑛}\right)+\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\\ & =\left({𝑢}_1+{𝑣}_1,{𝑢}_2+{𝑣}_2,\dots ,{𝑢}_{𝑛}+{𝑣}_{𝑛}\right)\\ & =\left({𝑣}_1+{𝑢}_1,{𝑣}_2+{𝑢}_2,\dots ,{𝑣}_{𝑛}+{𝑢}_{𝑛}\right)\\ & =\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)+\left({𝑢}_1,{𝑢}_2,\dots ,{𝑢}_{𝑛}\right)\\ & =\mathrm{𝐯}+\mathrm{𝐮}.\end{array}$$

3. Existence of vector additive identity. Choose ${\mathrm{𝟎}}_{𝑉}=\left(0_{𝐹},0_{𝐹},\dots ,0_{𝐹}\right)$, where $0_{𝐹}$ is the additive identity of $𝐹$. For any vector $\mathrm{𝐯}=\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\in {𝐹}^{𝑛}$, we have:

   $$\begin{array}{cc}\mathrm{𝐯}+{\mathrm{𝟎}}_{𝑉}& =\left({𝑣}_1+0_{𝐹},{𝑣}_2+0_{𝐹},\dots ,{𝑣}_{𝑛}+0_{𝐹}\right)=\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)=\mathrm{𝐯}.\end{array}$$

4. Existence of vector additive inverses. For any vector $\mathrm{𝐯}=\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\in {𝐹}^{𝑛}$, choose $-\mathrm{𝐯}=\left(-{𝑣}_1,-{𝑣}_2,\dots ,-{𝑣}_{𝑛}\right)$, where $-{𝑣}_{𝑖}$ is the additive inverse of ${𝑣}_{𝑖}$ in $𝐹$. Then:

   $$\mathrm{𝐯}+\left(-\mathrm{𝐯}\right)=\left({𝑣}_1+\left(-{𝑣}_1\right),{𝑣}_2+\left(-{𝑣}_2\right),\dots ,{𝑣}_{𝑛}+\left(-{𝑣}_{𝑛}\right)\right)=\left(0_{𝐹},0_{𝐹},\dots ,0_{𝐹}\right)={\mathrm{𝟎}}_{𝑉}.$$

5. Compatibility of field multiplicative identity. For any vector $\mathrm{𝐯}=\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\in {𝐹}^{𝑛}$:

   $$\begin{array}{cc}{1}_{𝐹}\mathrm{𝐯}& =\left(1_{𝐹}{𝑣}_1,1_{𝐹}{𝑣}_2,\dots ,1_{𝐹}{𝑣}_{𝑛}\right)=\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)=\mathrm{𝐯}.\end{array}$$

6. Distributivity of scalar multiplication over vector addition. For any scalar $𝑎\in 𝐹$ and any vectors $\mathrm{𝐮}=\left({𝑢}_1,{𝑢}_2,\dots ,{𝑢}_{𝑛}\right),\mathrm{𝐯}=\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\in {𝐹}^{𝑛}$:

   $$\begin{array}{cc}𝑎\left(\mathrm{𝐮}+\mathrm{𝐯}\right)& =𝑎\left(\left({𝑢}_1,{𝑢}_2,\dots ,{𝑢}_{𝑛}\right)+\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\right)\\ & =𝑎\left({𝑢}_1+{𝑣}_1,{𝑢}_2+{𝑣}_2,\dots ,{𝑢}_{𝑛}+{𝑣}_{𝑛}\right)\\ & =\left(𝑎\left({𝑢}_1+{𝑣}_1\right),𝑎\left({𝑢}_2+{𝑣}_2\right),\dots ,𝑎\left({𝑢}_{𝑛}+{𝑣}_{𝑛}\right)\right)\\ & =\left(𝑎{𝑢}_1+𝑎{𝑣}_1,𝑎{𝑢}_2+𝑎{𝑣}_2,\dots ,𝑎{𝑢}_{𝑛}+𝑎{𝑣}_{𝑛}\right)\\ & =\left(𝑎{𝑢}_1,𝑎{𝑢}_2,\dots ,𝑎{𝑢}_{𝑛}\right)+\left(𝑎{𝑣}_1,𝑎{𝑣}_2,\dots ,𝑎{𝑣}_{𝑛}\right)\\ & =𝑎\left({𝑢}_1,{𝑢}_2,\dots ,{𝑢}_{𝑛}\right)+𝑎\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\\ & =𝑎\mathrm{𝐮}+𝑎\mathrm{𝐯}.\end{array}$$

7. Distributivity of scalar multiplication over scalar addition. For any scalars $𝑎,𝑏\in 𝐹$ and any vector $\mathrm{𝐯}=\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\in {𝐹}^{𝑛}$:

   $$\begin{array}{cc}\left(𝑎+𝑏\right)\mathrm{𝐯}& =\left(𝑎+𝑏\right)\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\\ & =\left(\left(𝑎+𝑏\right){𝑣}_1,\left(𝑎+𝑏\right){𝑣}_2,\dots ,\left(𝑎+𝑏\right){𝑣}_{𝑛}\right)\\ & =\left(𝑎{𝑣}_1+𝑏{𝑣}_1,𝑎{𝑣}_2+𝑏{𝑣}_2,\dots ,𝑎{𝑣}_{𝑛}+𝑏{𝑣}_{𝑛}\right)\\ & =\left(𝑎{𝑣}_1,𝑎{𝑣}_2,\dots ,𝑎{𝑣}_{𝑛}\right)+\left(𝑏{𝑣}_1,𝑏{𝑣}_2,\dots ,𝑏{𝑣}_{𝑛}\right)\\ & =𝑎\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)+𝑏\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\\ & =𝑎\mathrm{𝐯}+𝑏\mathrm{𝐯}.\end{array}$$

8. Compatibility of scalar and field multiplication. For any scalars $𝑎,𝑏\in 𝐹$ and any vector $\mathrm{𝐯}=\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\in {𝐹}^{𝑛}$:

   $$\begin{array}{cc}\left(𝑎𝑏\right)\mathrm{𝐯}& =\left(𝑎𝑏\right)\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\\ & =\left(\left(𝑎𝑏\right){𝑣}_1,\left(𝑎𝑏\right){𝑣}_2,\dots ,\left(𝑎𝑏\right){𝑣}_{𝑛}\right)\\ & =\left(𝑎\left(𝑏{𝑣}_1\right),𝑎\left(𝑏{𝑣}_2\right),\dots ,𝑎\left(𝑏{𝑣}_{𝑛}\right)\right)\\ & =𝑎\left(𝑏{𝑣}_1,𝑏{𝑣}_2,\dots ,𝑏{𝑣}_{𝑛}\right)\\ & =𝑎\left(𝑏\left({𝑣}_1,{𝑣}_2,\dots ,{𝑣}_{𝑛}\right)\right)\\ & =𝑎\left(𝑏\mathrm{𝐯}\right).\end{array}$$

Suppose $𝐻$ is a subset of $𝑉$. Let $𝑝$ be the property that $𝐻$ is a subspace of $𝑉$, and let $𝑞$ be the property that $𝐻$ satisfies the three conditions listed in Theorem 7.

$𝒑\Rightarrow 𝒒$. Assume $𝐻$ is a subspace of $𝑉$. The vector additive identity of $𝑉$ is unique. Since $𝐻$ is a subspace of $𝑉$, its vector additive identity must be the same as that of $𝑉$, so ${\mathrm{𝟎}}_{𝑉}\in 𝐻$. $𝐻$ is a vector space under the same operations as $𝑉$, so $𝐻$ is closed under vector addition and scalar multiplication. Therefore, $𝑞$ is true.

$𝒒\Rightarrow 𝒑$. Assume that ${\mathrm{𝟎}}_{𝑉}\in 𝐻$, $𝐻$ is closed under vector addition, and $𝐻$ is closed under scalar multiplication. Then, axiom (3) is satisfied, as well as the requirement that ${+}_{𝑉}:𝐻\times 𝐻\to 𝐻$ and ${\cdot }_{𝑉}:𝐹\times 𝐻\to 𝐻$. Since $𝐻\subseteq 𝑉$, the axioms of vector spaces that involve only elements of $𝐻$ and the operations ${+}_{𝑉}$ and ${\cdot }_{𝑉}$ (namely axioms 1, 2, 5, 6, 7, and 8) must also hold for $𝐻$ since they hold for all elements of $𝑉$. Finally, axiom (4) is satisfied since for any $\mathrm{𝐯}\in 𝐻$, we know that $\left(-1_{𝐹}\right)\mathrm{𝐯}\in 𝐻$ by closure of $𝐻$ under scalar multiplication, and $\left(-1_{𝐹}\right)\mathrm{𝐯}=-\mathrm{𝐯}$, so $-\mathrm{𝐯}\in 𝐻$ by Theorem 5. Thus, $𝐻$ satisfies all vector space axioms under the same operations as $𝑉$, and $𝐻$ is a subset of $𝑉$, so $𝐻$ is a subspace of $𝑉$. Therefore, $𝑝$ is true.

Since $𝑝\Rightarrow 𝑞$ and $𝑞\Rightarrow 𝑝$, we have $𝑝\iff 𝑞$. ∎

Suppose $𝑉$ is a vector space over $𝐹$, and ${\mathrm{𝟎}}_{𝑉}$ is the vector additive identity of $𝑉$. Since ${\mathrm{𝟎}}_{𝑉}\in 𝑉$, it follows that $\left\{{\mathrm{𝟎}}_{𝑉}\right\}\subseteq 𝑉$.

1. Existence of additive identity. ${\mathrm{𝟎}}_{𝑉}\in \left\{{\mathrm{𝟎}}_{𝑉}\right\}.$
2. Closure under vector addition. For any $\mathrm{𝐮},\mathrm{𝐯}\in \left\{{\mathrm{𝟎}}_{𝑉}\right\}$, we have $\mathrm{𝐮}=\mathrm{𝐯}={\mathrm{𝟎}}_{𝑉}$, so $\mathrm{𝐮}+\mathrm{𝐯}={\mathrm{𝟎}}_{𝑉}+{\mathrm{𝟎}}_{𝑉}={\mathrm{𝟎}}_{𝑉}\in \left\{{\mathrm{𝟎}}_{𝑉}\right\}$.
3. Closure under scalar multiplication. For any scalar $𝑎\in 𝐹$ and any vector $\mathrm{𝐯}\in \left\{{\mathrm{𝟎}}_{𝑉}\right\}$, we have $\mathrm{𝐯}={\mathrm{𝟎}}_{𝑉}$, so $𝑎\mathrm{𝐯}=𝑎{\mathrm{𝟎}}_{𝑉}={\mathrm{𝟎}}_{𝑉}\in \left\{{\mathrm{𝟎}}_{𝑉}\right\}$.

By Theorem 7, $\left\{{\mathrm{𝟎}}_{𝑉}\right\}$ is a subspace of $𝑉$. ∎