Linear transformations

Objective. Find kernels, images and matrix representations.

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Learn: the key idea

Linear transformations are functions T: V → W between vector spaces satisfying T(u + v) = T(u) + T(v) and T(cv) = cT(v). Every such map between finite-dimensional spaces can be represented by a matrix once bases are fixed, with the kernel ker(T) = {v : T(v) = 0} and image im(T) = {T(v) : v ∈ V} describing its structure. For T: ℝ² → ℝ² given by T(x,y) = (x+y, 2x+2y), the matrix is [[1,1],[2,2]], and ker(T) consists of vectors with x = −y, a one-dimensional subspace. Understanding kernel and image reveals whether T is injective or surjective.

Worked example

Find the kernel of the linear transformation T(x, y, z) = (x − y, y − z) from ℝ³ to ℝ².

1 $Set T(x,y,z) = (0,0), giving x - y = 0 and y - z = 0.$
2 $From these, x = y and y = z, so x = y = z.$
3 $The kernel consists of all vectors of the form (t, t, t) for t \in ℝ.$
4 $This is a one-dimensional subspace spanned by (1, 1, 1).$

Answer: ker(T) = span{(1, 1, 1)}

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Common mistake. Students assume any map given by a formula with matrices and vectors is automatically linear; T(x,y) = (x+1, y) fails linearity since T(0,0) = (1,0) ≠ (0,0), so it is merely affine, not linear.

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