1. Definition of a Linear Transformation
Linear Transformation: T : R^n → R^m is linear if T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors u, v and scalars c.
Equivalently, T(c1*u + c2*v) = c1*T(u) + c2*T(v). A linear transformation always maps the zero vector to the zero vector: T(0) = 0.
Common examples: rotations, reflections, projections, scaling, and shearing. Non-examples: any function with a constant term (like T(x) = x + 1) or nonlinear operations (like T(x,y) = xy).
2. The Standard Matrix
A = [T(e1) | T(e2) | ... | T(en)]
Standard Matrix: Every linear transformation T : R^n → R^m can be written as T(x) = Ax for a unique m x n matrix A. The columns of A are the images of the standard basis vectors.
To find A: compute T(e1), T(e2), ..., T(en) and use them as columns.
3. Important Standard Transformations
| Transformation | Standard Matrix (R^2) |
|---|
| Rotation by angle theta | [cos(theta) -sin(theta); sin(theta) cos(theta)] |
| Reflection over x-axis | [1 0; 0 -1] |
| Reflection over y-axis | [-1 0; 0 1] |
| Projection onto x-axis | [1 0; 0 0] |
| Scaling by factor k | [k 0; 0 k] |
4. Kernel and Range
Kernel (Null Space): ker(T) = {x in R^n : T(x) = 0}. The set of all inputs that map to zero.
Range (Image): range(T) = {T(x) : x in R^n} = column space of A.
T is one-to-one if and only if ker(T) = {0}. T is onto if and only if range(T) = R^m.
To find the kernel: solve Ax = 0 (null space). To find the range: find the column space of A (identify pivot columns).
5. Rank-Nullity Theorem
rank(T) + nullity(T) = dim(domain) = n
Rank: dim(range(T)) = number of pivot columns. Nullity: dim(ker(T)) = number of free variables.
The dimensions of the kernel and range are complementary -- they always add up to the dimension of the domain. This constrains when T can be one-to-one or onto.
| Condition | Means |
|---|
| Nullity = 0 | T is one-to-one |
| Rank = dim(codomain) | T is onto |
| n > m (more columns than rows) | T cannot be one-to-one |
| n < m (fewer columns than rows) | T cannot be onto |