What's a Vector (Space) / Subspace?

A vector space is a set $V$ of vectors, coupled with two operations $+$ and $\cdot$, called vector addition and scalar multiplication, respectively, that satisfy the following ten properties:

1. (Closure under vector addition) If $u, v\in V$, then $u+v\in V$.

2. (Commutativity of vector addition) For any two vectors $u, v\in V$, $u+v=v+u$.

3. (Associativity of vector addition) For any three vectors $u, v, w\in V$, $(u+v)+w = u+(v+w)$.

4. (Existence of an additive identity) There exists a zero vector $0\in V$ such that $0+u=u$ for all vectors $u\in V$.

5. (Existence of an additive inverse) For every vector $u\in V$, there exists vector $-u\in V$, called the additive inverse of $u$, such that $(-u)+u=0$.

6. (Closure under scalar multiplication) If $u\in V$, then $\alpha\cdot u\in V$ for all scalars $\alpha$.

7. ("Associativity" of scalar multiplication) For any two scalars $\alpha$ and $\beta$ and any vector $u\in V$, $\alpha\cdot(\beta\cdot u) = (\alpha\beta)\cdot u$.

8. (Scalar multiplicative identity) For all $u\in V$, $1\cdot u = u$.

9. (Distributivity of scalar multiplication over scalar addition) For any two scalars $\alpha$ and $\beta$ and any vector $u\in V$, $(\alpha+\beta)\cdot u = \alpha\cdot u + \beta\cdot u$.

10. (Distributivity of scalar multiplication over vector addition) For any scalar $\alpha$ and any two vectors $u, v\in V$, $\alpha\cdot(u+v) = \alpha\cdot u + \alpha\cdot v$.

The set of scalars in question will depend on the specific vector space; thus, the set of scalars should be specified if not immediately obvious. The most common sets of scalars are $\mathbb{R}$ (for real vector spaces) and $\mathbb{C}$ (for complex vector spaces), but any scalar field can serve as the set of scalars for a vector space.

Note that "vectors" can denote any mathematical object; they do not need to be vectors in the standard sense. For example, the set of all real numbers, $\mathbb{R}$ is a perfectly good vector space, assuming that vector addition and scalar multiplication simply correspond to standard addition and multiplication for real numbers.

Although $10$ conditions might seem like a lot to check, they are usually fairly easy to verify, often following directly from commutative, associative, and other such properties known to be satisfied by well-known operations such as addition of real numbers. Moreover, remember that, if you want to show that a set is not a vector space, you need only show that at least one condition is not met.

A subspace $U$ of a vector space $V$ is a set $U\subset V$, coupled with the same vector addition and scalar multiplication operations as $V$, such that $U$ is also a vector space.

If you want to know whether a subset $U$ of a given vector space $V$ is a subspace, you do not need to check all $10$ conditions! Since $U\subset V$, any element of $U$ must also be an element of $V$. Therefore, since $V$ already satisfies all $10$ properties above, $U$ must also satisfy properties $2$ through $5$ and $7$ through $10$. It only remains to check properties:

a) Additive identity $0 \in U$.

b) Closure under vector addition: if $u,v \in U$ then $u+v \in U$.

c) Closure under scalar multiplication: if $u \in V$ then $cu \in V$ for $c$ is a scalar.

From this we can follow that $\{0\}$ is a smallest subspace of vector space $V$ on $\mathbf{R}$ and $V$ is the largest subspace of $V$.

Example: $\{ (x_1,x_2,0): x_1,x_2 \in \mathbf{R} \}$ is a subspace of $\mathbf{R}^3$.

What are subspaces of $\mathbf{R}^2$?