Linear algebra you will actually use

Linear algebra can feel like a collection of disconnected rules until you see the same geometric ideas reappear in code. For graphics and signal-processing work, these four operations are especially useful — and knowing *why* each one is there makes the notation readable.

1 · Basis and coordinates. A vector is a thing; its coordinates are a description of that thing in a chosen basis. Almost every "mysterious" transformation becomes obvious the moment you ask: what basis is this written in?

# 1) basis change is just multiplying by a matrix of the NEW basis vectors
#     same point, two descriptions:
B = np.array([[np.cos(0.6), -np.sin(0.6)],      # rotated basis
              [np.sin(0.6),  np.cos(0.6)]])
p_world = np.array([1.0, 0.0])
p_local = np.linalg.inv(B) @ p_world           # describe p in B's basis

# 2) projection: split a vector into "along b" + "across b"
b = np.array([2.0, 1.0])
along  = (p_world @ b) / (b @ b) * b           # the part parallel to b
across = p_world - along                       # the remainder

# 3) least squares: solve A x = y when there is no exact answer
#    (this is what "fitting a line/plane/model" actually is)
x_hat, *_ = np.linalg.lstsq(A, y, rcond=None)

# 4) SVD: orthogonal transform . nonnegative stretch . orthogonal transform.
#    Orthogonal transforms may include reflections. Use SVD to compress,
#    denoise, or find dominant directions in data.
U, S, Vt = np.linalg.svd(A)

2 · Projection. Splitting a vector into a part along a direction and a part across it. This is how dot products, shadows, lighting, least-squares fitting and PCA all work — the same operation wearing different names.

3 · Least squares. When a system has no exact solution (often because it is over-determined), minimise the squared residual. Geometrically, this projects the target vector onto the column space of `A`; the residual from that closest representable output is perpendicular to the column space. Curve fitting, camera calibration and linear regression all use this idea.

4 · SVD. Every real matrix can be written as `U Σ Vᵀ`: an orthogonal transform, an axis-aligned nonnegative stretch, then another orthogonal transform. In 2D or 3D, an orthogonal transform may include a reflection as well as a rotation. This decomposition lets you compress (discard tiny singular values), denoise and find dominant directions.