Two-Dimensional Linear Algebra

This is a reference document on Two-Dimensional Linear Algebra by Fengyang Wang. It covers a subset of the material of MATH 136 or MATH 146.

Vector Spaces and Subspaces

A vector space $V$ is a set of objects named vectors over a field $F$, along with binary operations $+: V×V\to V$ and $⋅: F×V\to V$, called addition and scalar multiplication respectively. The $\cdot$ multiplication symbol is frequently omitted.

A vector space must satisfy the following axioms:

  • Addition is associative

  • Addition is commutative

  • There is a zero vector and it is the additive identity

  • Each vector has an additive inverse

  • For all $a∈F$, $b∈F$, $\mathbf{v}∈V$, we have $(ab)\mathbf{v} = a(b\mathbf{v})$

  • For all $a∈F$, $\mathbf{u}∈V$, and $\mathbf{v}∈V$, we have $a(\mathbf{u}+\mathbf{v}) = a\mathbf{u} + a\mathbf{v}$

  • For all $a∈F$, $b∈F$, $\mathbf{v}∈V$, we have $(a+b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}$

Note that the closure and totality properties are implicit in our definition of binary operations.

A subspace $W$ of a vector space $V$ is a subset of vectors in $V$ that itself forms a vector space, inheriting the addition and scalar multiplication operations from $V$.

The Vector Space $\mathbf{R}^2$

A vector in $\mathbf{R}^2$ is a pair of numbers $(a, b)$ where $a∈\mathbf{R}$ and $b∈\mathbf{R}$. We define the following operations on vectors:

  • $(a, b) + (c, d) = (a + c, b + d)$

  • $t (a, b) = (ta, tb)$ where $t∈\mathbf{R}$

Typically we write vectors in boldface font; e.g. $\mathbf{v}∈\mathbf{R}^2$. We denote by $\mathbf{0}$ the vector $(0, 0)$.

Note $\mathbf{R}^2$ is an example of a vector space. Then a subspace of $\mathbf{R}^2$ is a set of vectors such that:

  • the sum of any two vectors in the set is also in the set;

  • any scalar multiples of vectors in the set are in the set;

  • $\mathbf{0}$ is in the set.

Frequently we will write the vector $(a, b)$ as a column vector

\[\begin{bmatrix} a \\ b \end{bmatrix}\]

for notational convenience.

Linear Transformations

A linear transformation $T: V \to V$ on vector space $V$ is a function satisfying that for any vectors $\mathbf{u}∈V$, $\mathbf{v}∈V$, and scalars $a∈F$, $b∈F$, then

\[T(a\mathbf{u} + b\mathbf{v}) = aT\mathbf{u} + bT\mathbf{v}\]

Note that we frequently omit the brackets when applying a linear transformation; that is, $T\mathbf{u}$ means $T(\mathbf{u})$.

It is a theorem that all linear transformations from $\mathbf{R}^2$ to itself are of the form

\[(x, y) ↦ (ax + by, cx + dy)\]

so we frequently represent these linear transformations in a more compact form, as matrices

\[\begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

where we define

\[\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}\]

2×2 Real Matrices

The 2×2 real matrices themselves form a vector space $\mathbf{R}^{2×2}$ under the typical elementwise operations $+$ and $⋅$. Since there are four free real parameters, we say that such a vector space is four-dimensional. (A proper understanding of dimension is not needed for this talk and so I will skip it.)

We define multiplication between two matrices (note that this is distinct from scalar multiplication) to be the composition of their linear transformations. We do this definition because we wish for $(AB)\mathbf{v} = A(B\mathbf{v})$. For 2×2 real matrices, we have the formula

\[\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}\]

Linear Combinations and Spanning Sets

Suppose we have some finite set of vectors $\{\mathbf{v}_1, \dots, \mathbf{v}_n\} ⊆ V$ from real vector space $V$. Then for any scalars $\{a_1, \dots, a_n\} ⊆ \mathbf{R}$, the vector

\[\mathbf{v} = a_1\mathbf{v}_1 + \dots + a_n\mathbf{v}_n\]

is said to be a linear combination of $\{\mathbf{v}_1, \dots, \mathbf{v}_n\}$. (We can have linear combinations of an infinite number of vectors, but there is some more subtlety and it is not needed for this talk.)

Suppose we have some set of vectors $S = \{\mathbf{v}_1, \dots, \mathbf{v}_n\}$ from vector space $V$. Then the set of all linear combinations of $S$ is a subspace of $V$, and we call it $\operatorname{span}{S}$. This subspace is the smallest subspace of $V$ that contains all vectors in $S$.