Linear transformations
A function that respects addition and scalar multiplication is determined by what it does to a basis, and that fits in a matrix.
01Functions that respect the structure
A linear transformation is a function between vector spaces that plays well with the two operations:
From the second condition, with k = 0, it follows that T(0) = 0: every linear transformation sends the zero vector to the zero vector. That rules out f(x) = x + 1 right away, which translates, and f(x) = x², which does not respect scaling. Rotations, scalings, reflections, projections, the derivative and the integral are linear.
A circuit made of resistors, capacitors, inductors and linear controlled sources is a linear transformation from input to output. That is why superposition holds: you turn off everything except one source, solve, and then add the results. And that is why a linear circuit driven with a sinusoid responds with a sinusoid of the same frequency: the amplitude and phase change, not the shape.
02Every linear transformation fits in a matrix
If you know what T does to the vectors of a basis, you already know what it does to any vector: you write it as a linear combination of the basis and apply linearity. Writing the images of the basis as columns gives the associated matrix:
The matrix depends on the chosen bases. The same transformation, seen in another basis, has another matrix: they are similar matrices, related by A′ = P−1AP. Choosing the right basis so that the matrix ends up as simple as possible is the goal of diagonalization.
03The transformations of the plane, one by one
| Transformation | Matrix | det | What it does to the plane |
|---|---|---|---|
| Rotation by angle φ | [[cos φ, −sin φ], [sin φ, cos φ]] | 1 | Turns everything without deforming it; preserves areas and angles |
| Scaling | [[k₁, 0], [0, k₂]] | k₁k₂ | Stretches each axis separately |
| Reflection across the x axis | [[1, 0], [0, −1]] | −1 | Mirrors; the negative sign indicates that it reverses orientation |
| Shear | [[1, k], [0, 1]] | 1 | Slides the horizontal layers; preserves area but deforms angles |
| Projection onto the x axis | [[1, 0], [0, 0]] | 0 | Flattens the plane onto a line: it loses a dimension and has no inverse |
The light figure is the original and the colored one is its image. The transformed grid shows what happens to the whole plane.
04Kernel, image and the rank-nullity theorem
The vectors that the transformation sends to zero: Ker(T) = {X : A·X = 0}. It is the solution space of the homogeneous system. Its dimension is called the nullity.
Everything the transformation can produce: Im(T) = {A·X}. It is spanned by the columns of A, and its dimension is the rank.
The projection of the plane onto the x axis has a kernel of dimension 1 —the whole y axis is flattened to the origin— and an image of dimension 1: 1 + 1 = 2 ✔. A transformation is injective exactly when its kernel is only the zero vector, and in that case, if it also goes from one space to another of the same dimension, it is an isomorphism: it has an inverse, and the matrix of the inverse is A−1.
05Composing is multiplying
Applying T first and then S is equivalent to multiplying the matrices, in the order in which they are applied, from right to left: the matrix of S ∘ T is AS · AT. That is the origin of the fact that matrix multiplication is not commutative: rotating and then reflecting is not the same as reflecting and then rotating.
It is exactly what happens with two stages of a circuit in cascade, which was seen in matrices: each stage is a linear transformation and connecting them is composing them.
06Where they show up in electronics
- Superposition. It is linearity, nothing more. It holds only in linear circuits: a diode or a transistor in large-signal operation breaks the property, and that is why you have to linearize around an operating point before using it.
- Filters. A filter is a linear operator on signals. Its “matrix”, in the frequency domain, is the transfer function, which multiplies each spectral component.
- Discrete Fourier transform. The N-point DFT is, literally, multiplication by an N × N matrix whose elements are roots of unity. The FFT is the algorithm that does that product in N log N operations instead of N².
- Graphics and CAD. Rotating, scaling and projecting a model on screen are chained linear transformations: a single matrix summarizes the whole chain.
07In the lab
Find the matrix of the transformation of R² that rotates by 90° and then doubles the x component. Apply it to three vectors and verify with the lab above. Check that composing in the reverse order gives a different matrix.
For A = [[1, 2], [2, 4]], find the kernel by solving A·X = 0, describe the image and verify the rank-nullity theorem. Explain what happens to the grid in the lab with that matrix.
In a circuit with two sources, measure the output voltage with each source separately (the other one turned off: the voltage source short-circuited, the current source open-circuited) and then with both together. Check that the sum matches. Repeat adding a diode and explain why it no longer holds.
08Common mistakes
- Believing that every function “with a simple formula” is linear. f(x) = x + 1 is not: it does not send zero to zero.
- Building the matrix with the images of the basis as rows instead of columns.
- Reversing the order when composing: the matrix of S ∘ T is AS·AT.
- Confusing kernel with image, or their dimensions.
- Applying superposition to a circuit with nonlinear elements.
- Forgetting that the matrix depends on the basis: two different matrices can be the same transformation.
09Self-assessment
Is T(x, y) = (2x − y, 0) linear?
Yes. Its matrix is [[2, −1], [0, 0]]: it satisfies both conditions and sends zero to zero.
What is the matrix of the 90° counterclockwise rotation?
[[0, −1], [1, 0]]: it sends î to (0,1) and ĵ to (−1,0).
If A is 3 × 3 with rank 2, what is the dimension of the kernel?
3 − 2 = 1: a line of vectors ends up at the origin.
When does a transformation from Rn to Rn have an inverse?
When its kernel is only the zero vector, which is equivalent to det(A) ≠ 0 and to rank n.
What does it mean for the determinant of the matrix to be −1?
That it preserves areas but reverses orientation: there is a reflection.
Why does superposition not hold with a transistor in switching operation?
Because its behavior is not linear: doubling the input does not double the output, and the response to the sum of two inputs is not the sum of the responses.
10Further reading
- David C. Lay. Linear Algebra and Its Applications. 5th ed. (Spanish edition, “Álgebra lineal y sus aplicaciones”), Pearson, 2016. Develops linear transformations from the matrix, with many figures of the transformed plane.
- Stanley I. Grossman. Álgebra lineal. 6th ed., McGraw-Hill, 2008 (in Spanish). Kernel, image, the rank-nullity theorem and similar matrices, with complete proofs.
- Gilbert Strang. Linear Algebra and Its Applications. 4th ed. (Spanish edition, “Álgebra lineal y sus aplicaciones”), Cengage Learning, 2007. The “the matrix is the transformation” point of view, and the chapter on the Fourier transform as a change of basis.