Linear Transformations

Definition Vectors in $\mathbb{R}^n$ are columns, and on this page a tuple such as $(x, y)$ written inline stands for the column with those entries. A transformation, or map, $T$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ is a rule that assigns to each vector $\mathbf{x}$ in $\mathbb{R}^n$ exactly one vector $T(\mathbf{x})$ in $\mathbb{R}^m$. The space $\mathbb{R}^n$ of allowed inputs is the domain of $T$, and the space $\mathbb{R}^m$ where the outputs live is its codomain. The map $T$ is a linear transformation if it has two properties. The first, additivity, requires

for all vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^n$. The second, homogeneity, requires

for every vector $\mathbf{u}$ in $\mathbb{R}^n$ and every real number $c$. Together the two properties are equivalent to the single condition $T(a\mathbf{u} + b\mathbf{v}) = aT(\mathbf{u}) + bT(\mathbf{v})$ for all $\mathbf{u}, \mathbf{v}$ in $\mathbb{R}^n$ and all real $a, b$. Additivity splits the left side into $T(a\mathbf{u}) + T(b\mathbf{v})$ and homogeneity pulls out $a$ and $b$. Conversely, taking $a = b = 1$ gives additivity, and taking $b = 0$ gives homogeneity. Repeating the condition shows that a linear map sends every linear combination $c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k$ to $c_1T(\mathbf{v}_1) + \cdots + c_kT(\mathbf{v}_k)$, with the same coefficients.

Definition A map $T$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ preserves zero when it sends the zero vector of $\mathbb{R}^n$ to the zero vector of $\mathbb{R}^m$,

Every linear transformation preserves zero. Homogeneity with $c = 0$ proves it, since $T(\mathbf{0}) = T(0\mathbf{u}) = 0\,T(\mathbf{u}) = \mathbf{0}$ for any $\mathbf{u}$ in $\mathbb{R}^n$. Consequently a map with $T(\mathbf{0}) \ne \mathbf{0}$ is not linear. Zero preservation is necessary for linearity but not sufficient, because a map that preserves zero can still break additivity or homogeneity.

Definition For $j = 1, \dots, n$, the standard unit vector $\mathbf{e}_j$ in $\mathbb{R}^n$ has entry $1$ in position $j$ and $0$ in every other position. In $\mathbb{R}^3$, for example, $\mathbf{e}_1 = (1, 0, 0)$, $\mathbf{e}_2 = (0, 1, 0)$ and $\mathbf{e}_3 = (0, 0, 1)$. The ordered list $(\mathbf{e}_1, \dots, \mathbf{e}_n)$ is called the standard basis of $\mathbb{R}^n$. Every vector $\mathbf{x} = (x_1, \dots, x_n)$ is the combination

whose coefficients are the entries of $\mathbf{x}$. If $T$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ is linear, applying additivity repeatedly and then homogeneity to each term gives

So a linear map is completely determined by the $n$ images $T(\mathbf{e}_1), \dots, T(\mathbf{e}_n)$, and two linear maps that agree on every $\mathbf{e}_j$ agree on every input.

Definition Let $T$ be a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$. Its standard matrix is the $m \times n$ matrix whose column $j$ is $T(\mathbf{e}_j)$,

Then $T(\mathbf{x}) = A\mathbf{x}$ for every $\mathbf{x}$ in $\mathbb{R}^n$, because $A\mathbf{x}$ is the weighted sum $x_1T(\mathbf{e}_1) + \cdots + x_nT(\mathbf{e}_n)$ of the columns. No other $m \times n$ matrix has this property, since multiplying a matrix by $\mathbf{e}_j$ returns its column $j$, which must therefore equal $T(\mathbf{e}_j)$. Conversely, for every real $m \times n$ matrix $A$ with columns $\mathbf{a}_1, \dots, \mathbf{a}_n$, the map $\mathbf{x} \mapsto A\mathbf{x}$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ is linear. Indeed

for all $\mathbf{u}, \mathbf{v}$ in $\mathbb{R}^n$ and real $c$, which follows from $A\mathbf{x} = x_1\mathbf{a}_1 + \cdots + x_n\mathbf{a}_n$ by splitting each coefficient $u_j + v_j$ into its two parts and by factoring $c$ out of every coefficient $cu_j$. So, with the standard coordinates fixed, the linear maps from $\mathbb{R}^n$ to $\mathbb{R}^m$ are exactly the maps $\mathbf{x} \mapsto A\mathbf{x}$, each given by exactly one matrix. The matrix need not be square.

Definition For a fixed vector $\mathbf{b}$ in $\mathbb{R}^n$, translation by $\mathbf{b}$ is the map $\mathbf{x} \mapsto \mathbf{x} + \mathbf{b}$ from $\mathbb{R}^n$ to $\mathbb{R}^n$. If $\mathbf{b} \ne \mathbf{0}$, this map is not linear. It sends $\mathbf{0}$ to $\mathbf{b} \ne \mathbf{0}$, and it also breaks additivity, since translating $\mathbf{u} + \mathbf{v}$ gives $\mathbf{u} + \mathbf{v} + \mathbf{b}$ while adding the translated vectors gives $\mathbf{u} + \mathbf{v} + 2\mathbf{b}$. Only translation by $\mathbf{0}$, which is the identity map, is linear.

Definition A map $T$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ is nonlinear if it is not linear, that is, if additivity fails for at least one pair of vectors $\mathbf{u}, \mathbf{v}$ or homogeneity fails for at least one vector $\mathbf{u}$ and one real number $c$. A single explicit failure proves that a map is nonlinear. On their own, finitely many successful checks never prove linearity, which needs an argument that covers every input, such as an identity in arbitrary coordinates or a matching matrix formula. Likewise the images $T(\mathbf{e}_1), \dots, T(\mathbf{e}_n)$ alone do not determine a map that is not known to be linear.