# 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

1. Closure under Addition: For any vectors $u,v\in V$ their sum $u+v$ is also in $V$.
2. Associativity of Addition: Addition is associative; $\left(u+v\right)+w=u+\left(v+w\right)$ for all $u,v,w\in V$.
3. Commutativity of Addition: $u+v=v+u$for all $u,v\in V$.
4. Existence of Additive Identity: There exists a vector $0$ in $V$ such that $v+0=v$ for all $v\in V$.
5. Existence of Additive Inverse: For every $v\in V$, there exists a vector $-v$ in $V$ such that $v+\left(-v\right)=0$.
6. Closure under Scalar Multiplication: For any scalar $c$ and vector $v$ in $V$, the scalar multiplication $c\cdot v$ is in $V$.
7. Distributivity of Scalar over Vector Addition: $c\cdot \left(u+v\right)=c\cdot u+c\cdot v$ for all scalars $c$ and vectors $u,v\in V$.
8. Distributivity of Scalars: $\left(c+d\right)\cdot v=c\cdot v+d\cdot v$ for all scalars $c,d$ and $v\in V$.
9. Compatibility of Scalar Multiplication with Field Multiplication: $c\cdot \left(d\cdot v\right)=\left(c\cdot d\right)\cdot v$ for all scalars $c,d$ and $v\in V$.
10. 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

1. Euclidean Space: The set of $n$-tuples of real numbers ${\mathbb{R}}^{n}$ equipped with standard addition and scalar multiplication.
2. Polynomial Space: The set of all polynomials of degree at most $n$ with coefficients from a field.
3. Function Space: The set of all real-valued functions on a given interval $\left[a,b\right]$ 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 $T\left(u+v\right)=T\left(u\right)+T\left(v\right)$ and $T\left(c\cdot v\right)=c\cdot T\left(v\right)$ for all $u,v\in V$ and scalars $c$.