# Vector Spaces

#### Definition

**Vector Space:** A vector space, denoted as $V$, is a set equipped with two operations: vector addition and scalar multiplication, satisfying specific properties.

#### Properties of Vector Spaces

**Closure under Addition:** For any vectors $u,v\in V$ their sum $u+v$ is also in $V$.
**Associativity of Addition:** Addition is associative; $(u+v)+w=u+(v+w)$ for all $u,v,w\in V$.
**Commutativity of Addition:** $u+v=v+u$for all $u,v\in V$.
**Existence of Additive Identity:** There exists a vector $0$ in $V$ such that $v+0=v$for all $v\in V$.
**Existence of Additive Inverse:** For every $v\in V$, there exists a vector $-v$in $V$ such that $v+(-v)=0$.
**Closure under Scalar Multiplication:** For any scalar $c$ and vector $v$ in $V$, the scalar multiplication $c\cdot v$ is in $V$.
**Distributivity of Scalar over Vector Addition:** $c\cdot (u+v)=c\cdot u+c\cdot v$ for all scalars $c$ and vectors $u,v\in V$.
**Distributivity of Scalars:** $(c+d)\cdot v=c\cdot v+d\cdot v$ for all scalars $c,d$ and $v\in V$.
**Compatibility of Scalar Multiplication with Field Multiplication:** $c\cdot (d\cdot v)=(c\cdot d)\cdot v$ for all scalars $c,d$ and $v\in V$.
**Scalar Multiplication Identity:** $1\cdot v=v$ for all $v\in V$, where $1$ is the multiplicative identity of the field.

#### Examples of Vector Spaces

**Euclidean Space:** The set of $n$-tuples of real numbers ${\mathbb{R}}^{n}$ equipped with standard addition and scalar multiplication.
**Polynomial Space:** The set of all polynomials of degree at most $n$ with coefficients from a field.
**Function Space:** The set of all real-valued functions on a given interval $[a,b]$ with operations defined pointwise.

#### Subspaces

**Subspace:** A subset $W$ of a vector space $V$ that is itself a vector space under the same operations (vector addition and scalar multiplication).

#### Basis and Dimension

**Basis:** A set of vectors that spans a vector space and is linearly independent.
**Dimension:** The number of vectors in a basis of a vector space; it represents the "size" of the vector space.

#### Linear Independence and Span

**Linear Independence:** A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others.
**Span:** The set of all possible linear combinations of a given set of vectors.

#### Linear Transformations

**Linear Transformation:** A function $T:V\to W$ between two vector spaces that preserves vector addition and scalar multiplication. It satisfies $T(u+v)=T(u)+T(v)$ and $T(c\cdot v)=c\cdot T(v)$ for all $u,v\in V$ and scalars $c$.