Linear Transformations

Precise definition

A map $T:V\to W$ is linear if $T(au+bv)=aT(u)+bT(v)$ for all vectors $u,v$ and scalars $a,b$. Equivalently, it preserves vector addition and scalar multiplication. Linearity forces $T(0)=0$.

Notation and mathematical language

After bases are chosen, the matrix $[T]_{C\leftarrow B}$ has column $j$ equal to $[T(b_j)]_C$. Then $[T(v)]_C=[T]_{C\leftarrow B}[v]_B$. Composition corresponds to matrix multiplication in compatible bases, with the rightmost map applied first.

Conceptual picture

A linear map is determined completely by its values on a basis because every vector is a unique linear combination. Its kernel measures collapsed directions and its range measures reachable outputs.

Conditions and key results

Preserving only addition over real spaces does not by itself establish full linearity without regularity assumptions; check both properties directly. Affine maps $T(x)=Ax+b$ with $b\ne0$ are not linear because they do not send zero to zero.

A reliable strategy

1. Declare domain, codomain, field, and operations.
2. Test additivity and homogeneity, or find one counterexample such as $T(0)\ne0$.
3. For a matrix, transform each ordered domain basis vector and place its codomain coordinates as a column.
4. Use kernel, range, and composition to interpret the map; verify results on a general vector.

Fully worked example

Interpretation and application

Linear transformations model rotations, projections, differentiation, encoding, and local approximations. The matrix is coordinate-dependent, while injectivity, rank, and eigenstructure represent the underlying map.

Common mistakes

Verification and reasonableness

• Check $T(0)=0$ as a quick necessary test.
• Apply the matrix to coordinates of a general or sample vector and compare with the original rule.
• Confirm dimensions: an $m\times n$ matrix represents a map from $n$ coordinate inputs to $m$ outputs.

Practice

1. Is $T(x)=3x$ linear?
2. Is $T(x)=3x+1$ linear?
3. What forms column $j$ of a transformation matrix?

Further deduction

A map between finite-dimensional spaces of the same dimension is injective iff surjective iff its matrix is invertible. For unequal dimensions, rank–nullity gives structural limits: a map from higher to lower dimension cannot be injective, and one from lower to higher cannot be surjective. These are impossibility results independent of the chosen coordinates.

Similarity expresses the same transformation in different bases. If $P$ converts new coordinates to old, then the new matrix is $P^{-1}AP$. This is not an arbitrary matrix product: it converts input coordinates, applies the old representation, then converts output coordinates back. The order becomes clear by following one vector.

An affine map $T(x)=Ax+b$ is linear exactly when $b=0$. Translation preserves lines and parallelism but fails $T(0)=0$ and usually fails additivity. Homogeneous coordinates can represent affine transformations with larger matrices, but that modelling device does not make the original map linear on the original vector space. Separating the matrix's linear action from the offset prevents incorrect kernel, eigenvector, or subspace arguments.

Related topics

• Kernel and Range
• Matrix Multiplication
• Basis