Linear Transformations: Matrices, Kernels and Images

A linear transformation is a function T between vector spaces respecting the two operations: T(u + v) = T(u) + T(v) and T(cv) = cT(v). Rotations, reflections, projections and scalings of the plane are linear, and so is differentiation on polynomials.

Linearity is powerful because the whole map is fixed by its values on a basis: know where the basis vectors go, and every other vector follows. That is why every linear map between finite-dimensional spaces is captured by a single matrix, turning geometry into computable arithmetic.

Key Concepts

• Linear transformation: T with T(u+v) = T(u)+T(v) and T(cv) = cT(v).
• Kernel ker(T): inputs sent to zero, a subspace measuring destroyed information.
• Image im(T): all outputs T(v), a subspace measuring reach.
• Matrix of T: with bases fixed, columns are the images of the domain basis vectors.
• Composition: S after T is linear, with matrix equal to the product of the matrices.

Formulas

• Linearity test: T(au + bv) = aT(u) + bT(v) for all scalars and vectors.
• Zero preserved: every linear map has T(0) = 0; a translation is not linear.
• Matrix action: with standard bases, T(x) = Ax by ordinary multiplication.
• Composition: matrix of S after T is (matrix of S)(matrix of T), in that order.
• Invertibility: between equal dimensions, invertible exactly when det is nonzero.

Worked Example

Let T(x, y) = (x + y, 2x minus y) on R squared. On the standard basis: T(1,0) = (1,2) and T(0,1) = (1,minus 1). By linearity, T(2,3) = 2(1,2) + 3(1,minus 1) = (5,1). Directly: (2+3, 4 minus 3) = (5,1), matching. The matrix has columns T(1,0), T(0,1), so A = [[1,1],[2,minus 1]], and A(2,3) = (5,1) agrees. Its determinant is (1)(minus 1) minus (1)(2) = minus 3, nonzero, so T is invertible.

Common Misconceptions

• Straight-looking maps are linear. T(x) = x + (1,0) preserves lines but T(0) is not zero, so it is affine, not linear.
• No squares in the formula means linear. T(x,y) = (x+1, y) has no squares yet is not linear.
• The matrix is basis-independent. Entries depend on the bases; changing bases conjugates the matrix.

Real-World Applications

• Graphics: every 2D/3D rotation, scale and projection in games is a linear transformation.
• Image processing: blur, sharpen and edge detection are linear maps on the space of images.
• Quantum mechanics: physical operations on states are linear (unitary) operators.

A linear transformation is fixed by its action on a basis, recorded column by column in its matrix. Test the two linearity equations, read kernel and image for what is destroyed and reached, and let matrix multiplication compute.