Linear algebra - vector spaces

Definitions and theorems regarding finite-dimensional vector spaces. Most of these are adapted from [1]. See 3Blue1Brown’s video for intuition.

Definition of linear algebra

Linear algebra is the study of linear maps on finite-dimensional vector spaces [1].

Vector space

A vector space is a set \(V\) which contains vectors in space where the properties of commutativity, associativity, additive identity, additive inverse, multiplicative identity, distributive properties hold (closed under these properties).

You should think of a vector space as a set of vectors with added structure/properties.

Linear combination

If \(a_{1}, \ldots, a_{m}\) are \(n\)-vectors, and \(\beta_{1}, \ldots, \beta_{m}\) are scalars, the \(n\)-vector

\[\beta_{1} a_{1}+\cdots+\beta_{m} a_{m}\]

is called a linear combination of the vectors \(a_{1}, \ldots, a_{n}.\) The scalars \(\beta_{1}, \ldots, \beta_{m}\) are called the coefficients of the linear combination.

You should think of linear combinations as adding and scaling vectors to get a different vector. In fact, In other words, a vector in a vector space can be defined by how much it scales the unit (basis) vectors (which is shown mathematically through a linear combination).

Basis

A basis of a vector space \(V\) is a set of vectors \(\left\{v_{1}, \ldots, v_{n}\right\}\) such that each vector \(v \in V\) can be written uniquely as a linear combination of \(v_{1}, \ldots, v_{n}.\)

Note: every basis of a vector space \(V\) has the same length.

Span

The set of all linear combinations of a list of vectors \(v_{1}, \ldots, v_{m}\) (i.e. vector space) in \(V\) is called the span of \(v_{1}, \ldots, v_{m}\), denoted \(\operatorname{span}(\left.v_{1}, \ldots, v_{m}\right).\) In other words, \(\operatorname{span}\left(v_{1}, \ldots, v_{m}\right)=\left\{a_{1} v_{1}+\cdots+a_{m} v_{m}: a_{1}, \ldots, a_{m} \in \mathbb{R}\right\}\)

In english, a span of a vector space is the set all of the possible vectors one can obtain from forming linear combinations with the basis vectors of that vector space.

Dimension

A dimension of a vector space \(V\) is the length of any basis of \(V\) [1].

Linear independence

All of this is nicely tied together with the most important definition:

A list \(v_{1}, \ldots, v_{m}\) of vectors in \(V\) is called linearly independent if the only choice of \(a_{1}, \ldots, a_{m} \in \mathbf{F}\) that makes \(a_{1} v_{1}+\cdots+a_{m} v_{m}\) equal 0 is \(a_{1}=\cdots=a_{m}=0\). If the list \(v_{1}, \ldots, v_{m}\) of vectors is not linearly independent, it is considered linearly dependent. In other words, a list of vectors is linearly independent if and only if all of them cannot be written as a linear combination of the others.

Geometrically (in \(\mathbb{R}^2\)), If we take all of the vectors in the list and plot them on the graph, the list is linearly independent if no two vectors sit on the same line.

Independence-dimension inequality

No list of vectors that is larger than the size of the smallest basis (another list of vectors) can be linearly independent (Length of linearly independent list \(\leq\) length of spanning list).

Linear map

Linear transformations (maps) are functions that map one vector space to another (where the transformation is linear). The following definition is adapted from [1].

Formally, a linear map from \(V\) to \(W\) is a function \(F: V \rightarrow W\) where the following properties hold (these properties are what make the transformation linear):

• Additivity: \(F(u+v)=F(u) + F(v) \text { for all } u, v \in V\)
• Homogeneity: \(F(\lambda v)=\lambda(F v) \text { for all } \lambda \in \mathbb{R} \text { and all } v \in V\)

[1] - Sheldon Jay Axler. Linear algebra done right. Vol. 2. Springer, 1997.