Mini Physics
Part of Mini EducationSwitch subject

Matrices and Linear Algebra (Physics Basics)

Learn how matrices represent linear transformations, solve linear systems, and use eigenvalues/eigenvectors (with worked examples).

  • University Physics Year 1
On this page

Linear algebra is the language of vectors, coordinate transformations, coupled systems, and “normal modes”. In Year 1, you mostly need: solving linear systems and seeing matrices as transformations.

Why this matters + quick links

This page is the “matrix toolkit” you reuse across UY1 physics:

Quick checks (systems)
  • Substitute your solution back into the original equations.
  • Units must match term-by-term (you can’t add meters to seconds).
  • If det(A) = 0, the system is either inconsistent (no solutions) or underdetermined (infinitely many solutions).

A common physics pattern: “solve for unknown components”

You’ll often solve systems like this when you resolve vectors into components or enforce constraints (force balance, circuit laws, geometry constraints). The structure is the same: unknowns are stacked into a vector vector x, equations become A vector x = vector b.

1) Matrices as linear transformations

A matrix maps vectors to vectors:

vector y = A vector x.

In 2D,

(y₁; y₂) = (a, b; c, d) (x₁; x₂) .

This viewpoint makes rotations and coordinate changes feel natural.

Columns-as-basis-vectors (a practical interpretation)

In 2D, if

A = (a, b; c, d),

then:

  • the first column (a,c) is where the basis vector (1,0) gets mapped,
  • the second column (b,d) is where the basis vector (0,1) gets mapped.

This is a fast way to sanity-check what a matrix is “doing” geometrically.

Worked example (rotation matrix + a built-in check)

A rotation by angle θ is represented by:

R(θ) = (cos θ, - sin θ; sin θ, cos θ) .

Apply it to vector x = (1,0):

R(θ) (1; 0) = (cos θ; sin θ),

which is exactly “a unit vector at angle θ”.

Quick check for a rotation matrix

A pure rotation should preserve lengths and angles. In matrix form that means:

R^T R = I and det(R) = +1.
Pitfall: active vs passive rotations

“Rotate the vector” and “rotate the coordinate axes” are not the same operation. The same physical situation can be described either way, but the matrix you write down may be R or R⁻¹ = R^T depending on the convention. This is why it helps to keep the geometry in mind (and to test with a simple vector like (1,0)).

2) Solving a linear system (worked example)

Consider:

2x + y = 1,; x-y = 3.

Write it as A vector x = vector b:

(2, 1; 1, -1) (x; y) = (1; 3) .

From x-y = 3, we have y = x-3. Substitute into 2x + y = 1:

2x + (x-3) = 1 ⇒ 3x = 4 ⇒ x = 4/3.

So

y = 4/3-3 = -5/3.
Uniqueness condition

A 2 × 2 system has a unique solution if the determinant is nonzero:

det(A) = ad-bc ≠ 0.

3) Eigenvalues and eigenvectors (the “special directions”)

An eigenvector vector v of a matrix A satisfies:

A vector v = λ vector v,

meaning A only scales vector v (does not change its direction). λ is the eigenvalue.

This shows up in physics as:

  • principal axes (diagonalizing quadratic forms),
  • normal modes of coupled oscillations,
  • stationary states (later, in quantum mechanics).

Worked example (normal modes of two coupled oscillators)

Take two identical masses m on a line, connected by three identical springs of constant k (wall–mass, mass–mass, mass–wall). Let x₁(t) and x₂(t) be small displacements from equilibrium.

The equations of motion can be written:

mvector x double dot = -k (2, -1; -1, 2) vector x, vector x = (x₁; x₂) .

Assume a harmonic solution (this is where complex exponentials are convenient):

vector x(t) = Re(vector Ae^(iω t)).

Then vector x double dot contributes a factor -ω², and you get an eigenvalue problem:

(2, -1; -1, 2) vector A = λ vector A, λ = mω²/k.

Now check two candidate eigenvectors:

  • In-phase motion vector A₁ = (1,1):

    (2, -1; -1, 2) (1; 1) = (1; 1) = 1 · (1; 1),

    so λ₁ = 1 and

    ω₁ = square root of (k/m) .

  • Out-of-phase motion vector A₂ = (1,-1):

    (2, -1; -1, 2) (1; -1) = (3; -3) = 3 · (1; -1),

    so λ₂ = 3 and

    ω₂ = square root of (3k/m) .

Interpretation: eigenvectors give the mode shapes (how the parts move together), and eigenvalues give the mode frequencies.

Key benefit (why eigenvectors are “physics solutions”)

In a good basis (the eigenvector basis), a coupled system becomes a set of independent single-oscillator equations. That’s the mathematical reason “normal modes” simplify real physics problems.

Pitfall: eigenvectors depend on your coordinates (but the physics doesn’t)

Mode shapes are usually drawn with simple vectors like (1,1) and (1,-1), but you can scale an eigenvector by any nonzero constant and it represents the same direction/mode. Frequencies (from eigenvalues) are the physically measurable part.

Jacobian connection (preview)

In multivariable calculus, the Jacobian matrix is literally “a matrix of partial derivatives”. It behaves like a linear map that approximates a nonlinear change of variables near a point. See: Partial derivatives.

  • Coordinate transformation under rotation

    Rotation matrices and how components change under a rotated axis.

    Open lesson

  • Vector Analysis

    Dot/cross products and vector operators used with linear algebra.

    Open lesson

  • Complex numbers

    Complex exponentials make harmonic motion and phasors easy to handle.

    Open lesson

  • Second-order differential equations

    Normal-mode solutions often use e^(iω t) and eigenvalues.

    Open lesson

  • Partial derivatives

    Jacobians are matrices of partial derivatives (a key link between calculus and linear algebra).

    Open lesson

Back to Mathematics for Undergraduate Physics