Vector Spaces II

Lecture 13

Minjae Park

Auburn University
MATH 2660 - Spring 2026

February 9, 2026

Abstraction

Abstraction allows us to recognize the same idea in different settings, so we do not have to relearn everything from scratch.

Numbers: \[ 2 + 3 = 5 \]
Vectors in \(\mathbb{R}^2\): \[ (1,2) + (3,4) = (4,6) \]
Matrices: \[ \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \]

In all of these examples:

The objects look different, but the rules are the same. These rules are called the axioms of addition.

If \[ u + v = u + w, \] then \[ v = w. \]

This holds for any objects satisfying the axioms of addition.

Abstraction captures the essential nature of a concept (for example, what “addition” really means).
Axioms act like an interface: they guarantee that certain rules always work (similar to checking whether a charging port is USB-C before plugging in a cable).
To understand an abstract structure (like a vector space), it helps to master one concrete example first (e.g., once you know how to use a USB-C charger for your phone, you can confidently charge almost any device with the same interface).
Once you understand \(\mathbb{R}^n\), most properties extend naturally to abstract vector spaces. In almost all cases, you can think of vector spaces as generalizations of Euclidean space.

A vector space is a safe place for linear combinations.

The Euclidean vector space \(\mathbb{R}^n\) (of dimension \(n\)) is the most fundamental example of a vector space.
Each vector is written using \(n\) coordinates, \(\langle x_1,\dots,x_n \rangle\), and vector operations are performed coordinate by coordinate.
Most of your geometric and algebraic intuition about vectors should come from this space.
Other examples of vector spaces can be understood by comparing them to \(\mathbb{R}^n\).

Consider \(f(x)=x^3+2x^2-1\) and \(g(x)=-2x^3+3\).
Compute \(3f(x)-2g(x)\):
- \(3f(x)=3x^3+6x^2-3\)
- \(-2g(x)=4x^3-6\)
- So \[ 3f(x)-2g(x)=7x^3+6x^2-9. \]
We combine coefficients term by term.
This is exactly how we compute linear combinations in \(\mathbb{R}^3\).
Compare with \(\vec u=\langle 1,2,-1 \rangle\), \(\vec v=\langle -2,0,3 \rangle\) and \(3\vec u-2\vec v\).

Consider \[ A=\begin{bmatrix}1&0\\1&1\end{bmatrix}, \qquad B=\begin{bmatrix}0&-1\\1&0\end{bmatrix}. \]
Compute \(2A+B\) entry by entry.
This is the same as linear combinations in \(\mathbb{R}^4\).
Compare with \(2\vec u +\vec v\) where \(\vec u=\langle 1,0,1,1 \rangle, \vec v=\langle 0,-1,1,0 \rangle\).

Consider real-valued functions \(f(x)\) and \(g(x)\).
A function can be viewed as a “collection of numbers” indexed by \(x\).
Linear combinations are computed pointwise: \[ (f-g)(x)=f(x)-g(x). \]
So the combination happens “index by index.”
In this sense, a function space behaves like an infinite-dimensional vector space with infintely many coordinates (indexed by \(x\)).

There are many natural subspaces inside the vector spaces we have seen.

Let \(\mathcal P_k\) be the set of polynomials of degree at most \(k\). Then \(\mathcal P_k\) is a vector space, and if \(m \le n\), we have \(\mathcal P_m \subseteq \mathcal P_n\), so \(\mathcal P_m\) is a subspace of \(\mathcal P_n\).
Let \(M_{n\times n}\) be the set of all \(n\times n\) matrices, \(L_{n\times n}\) the set of lower triangular matrices, and \(D_{n\times n}\) the set of diagonal matrices. Then \[ D_{n\times n} \subseteq L_{n\times n} \subseteq M_{n\times n}, \] so each is a subspace of the next.
The set of differentiable real-valued functions is a subspace of the set of continuous functions, which is itself a subspace of the set of all real-valued functions.

Coordinates such as \((1,2,-1)\) only make sense after we choose coordinate directions.
If vectors \(\vec v_1,\vec v_2,\vec v_3\) generate a 3-dimensional grid, then \[ (1,2,-1) \leftrightarrow \vec v_1+2\vec v_2-\vec v_3. \] So the coordinates describe how much of each direction we use.
In the vector space of polynomials, natural “coordinate directions” are \(1,\; x,\; x^2,\;x^3,\; \dots\)
In the subspace of polynomials of degree \(\le 2\), \[ (1,2,-1) \leftrightarrow 1+2x-x^2. \]
Thus \(\{1,x,x^2\}\) forms a spanning set for the space of polynomials of degree \(\le 2\).

Consider the space of \(2\times2\) lower triangular matrices: \[ \begin{bmatrix} a & 0\\ b & c \end{bmatrix}. \]
This space requires three parameters \((a,b,c)\).
Natural “coordinate directions” are \[ \begin{bmatrix}1&0\\0&0\end{bmatrix},\quad \begin{bmatrix}0&0\\1&0\end{bmatrix},\quad \begin{bmatrix}0&0\\0&1\end{bmatrix}. \]
Every lower triangular matrix is a linear combination of these three.

Most vector spaces contain infinitely many elements, yet they are generated (spanned) by finitely many vectors.
For example, \(\mathbb{R}^n\) is spanned by \(\{\vec e_1,\dots,\vec e_n\}\), and the space of \(n\times n\) matrices is spanned by \(n^2\) matrices, each having a single \(1\) in one entry and \(0\) elsewhere.
In this sense, the entire structure of a vector space is encoded in its spanning sets. They play the role of a “coordinate system,” since each vector can be written as a linear combination of these directions.
Subspaces arise naturally as the spans of subsets of a spanning set.

The set \[ \{(x_1,\dots,x_k,0,\dots,0): x_i\in\mathbb R\} \] is a subspace of \(\mathbb{R}^n\).
The whole space \(\mathbb{R}^n\) is spanned by \(\{\vec e_1,\dots,\vec e_n\}\).
The subspace above is spanned by the subset \(\{\vec e_1,\dots,\vec e_k\}\).
In \(\mathbb{R}^n\), the span of vectors forms a line, a plane, or more generally a \(k\)-dimensional subspace through the origin.
Intuitively, \(k\) independent vectors generate a \(k\)-dimensional grid.
The span must pass through the origin because the zero vector (obtained by taking all coefficients \(0\)) is always included

Let \(S=\{\langle 1,1 \rangle, \langle 2,2 \rangle\}\subset \mathbb{R}^2\). What is \(\mathop{\mathrm{span}}(S)\)?
Geometrically, we try to build a “grid” using \(\langle 1,1 \rangle\) and \(\langle 2,2 \rangle\). Normally, two vectors in \(\mathbb{R}^2\) generate a 2-dimensional grid (a plane).
However, these two vectors point in the same direction, so they lie on the same line.
Therefore, the two vectors only form a 1-dimensional grid (a line through the origin), and one vector is redundant.
Next class, we will learn how to detect such redundancy in general.
The key idea: if we have \(k\) non-redundant vectors (called a basis), they generate a \(k\)-dimensional space. The dimension of a vector space is the size of a minimal spanning set.