Vector Spaces

What is a vector space over a field \mathbb{F}?

A vector space is a set V equipped with two operation:

1. Vector addition: V\times V\rightarrow V

2. Scalar multiplication: \mathbb{F}\times V \rightarrow V

satisfying 8 axioms (closure, associativity, identity etc.). The elements of V are called vectors, and the scalar come from the field \mathbb{F}.

What are the 8 axioms of a vector space?

For all u,v,w \in V and \alpha, \beta\in\mathbb{F}:

1. u+v=v+u (commutativity)

2. (u+v)+w=u+(v+w) (associativity)

3. \exists 0\in V such that v+0=v (zero vector)

4. \exists(-v)\in V such that v+(-v)=0 (additive inverse)

5. 1\cdot v = v

6. \alpha(\beta v)=(\alpha\beta)v

7. (\alpha+\beta)v=\alpha v + \beta v

8. \alpha(u+v)=\alpha u+\alpha v

What are some common examples of vector spaces?

• \mathbb{F}^n: n-tuples of scalars

• Mat_{m\times n}(\mathbb{F}): all m\times n matrices

• \mathbb{F}[x]: polynomials with coefficients in \mathbb{F}

• C([a,b]): continuous functions on [a,b]

What s a subspace of a vector space V?

A subspace W\subseteq V is a subset that is itself a vector space under the same operations. It must satisfy:

• Contains zero vector

• Closed under addition

• Closed under scalar multiplication

What is the span of a set of vectors?

The span of a set S=\{v_1,…v_k\} is the set of all linear combinations of its elements Sp(S)=\{\sum^k_{i=1}\alpha_iv_i|\alpha_i\in\mathbb{F}\}. It is the smallest subspace containing all v_i.

What is the dimension of a vector space?

The number of vectors in any basis of the space. All bases have the same number of elements.

What is a basis of a vector space V?

A set \{v_1, …, v_n\}\subseteq V is a basis if it:

• Spans V, and

• Is linearly independent

Every vector in V can be written uniquely as a linear combination of the basis vectors.

Any two bases of a finite-dimensional vector space have the same number of elements, True or False?

True. This number is the dimension of the space. It ensures uniqueness of dimension.

What is the zero vector space?

The space \{0\} consisting only of the zero vector. It’s dimension is 0 and its only basis is the empty set.

What is the standard basis of \mathbb{F}^n?

e_1=(1,0,…,0),e_2=(0,1,0,…,0),…,e_n=(0,…,0,1). These vectors span \mathbb{F}^n and are linearly independent.

What is the coordinate vector is v in basis B?

If B=\{b_1,…,b_n\} and v=\sum\alpha_ib_i, then the coordinate vector is [v]_B=\left[\begin{array}{c} \alpha_i \\ \vdots \\ \alpha_n\end{array}\right].

What is a linearly dependent set of vectors?

A set \{v_1,…v_k\} is linearly dependent if there exists scalars (not all zero) such that \sum\alpha_iv_i=0 i.e. one vector is a linear combination of the others.

Extension and Reduction Principle

• Any linearly independent set in a finite-dimensional space can be extended to a basis

• Any spanning set can be reduced to a basis