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Kapittel 1 · Vektorrom
Dimension: How Many Free Directions?
Dimension as a count of linearly independent directions — a single arrow gives a line (dim 1), a second non-collinear arrow sweeps out a plane (dim 2), and a third arrow off the plane fills three-dimensional space (dim 3).
Linear Combinations: Mix Two Arrows
A linear combination is the recipe a·v₁ + b·v₂ with two scalar dials. Fix b at zero and sweep a — the output point glides along a line through the origin. Sweep both dials at once and the output paints out a region of the plane. Two non-parallel arrows span all of ℝ².
Span and Linear Dependence
Span of two non-parallel 2D arrows is a parallelogram (the whole plane); a third arrow inside the span is linearly dependent; a third arrow off the plane lifts the span to all of ℝ³, demonstrating linear independence.
Subspaces: Why the Origin Matters
A subspace must contain the zero vector, be closed under addition, and be closed under scalar multiplication. A line through the origin passes all three tests. A line that misses the origin fails closure under addition — the parallelogram-rule sum of two of its vectors lands off the line, and the origin is not on the line either.
Kapittel 2 · Lineærtransformasjoner
A Matrix Is a Function: T(v) = Av
A matrix-vector multiplication is a linear transformation. Walk through input v, the matrix machine A, the row-by-column dot product, and the output Av. Every linear T from ℝⁿ to ℝⁿ is multiplication by a matrix.
Change of Basis: P Converts the Reading
Change-of-basis matrix — the same vector has different coordinate readings in two bases. Stack the new basis vectors as columns of P; then P times the new-basis coordinates returns the standard-basis coordinates of the same vector.
Linear Transformations: A Matrix Moves the Grid
Linear transformation of the plane — every 2×2 matrix bends the standard grid in exactly one way, and its two columns are the images of the basis vectors e₁ and e₂. Rotation, scaling and shear are special cases of the same machine, closed off with a hero outro.
Rank, Nullity, and a Squashed Plane
Rank-nullity by example — a singular 2×2 matrix squashes the whole plane down to one line. The image is that surviving line (rank 1); the kernel is the direction that collapsed onto the origin (nullity 1). Together they account for the dimension of the domain.
Same Vector, Two Grids
Change of basis — the same arrow has different coordinates depending on the grid it is measured against. Standard grid reads our vector as (1, 3); a tilted basis B = {b₁, b₂} reads the very same vector as (2, 1).
The Determinant as Area
A 2×2 matrix maps the unit square to a parallelogram whose signed area is det A = ad − bc. A continuous morph from identity to A shows the deformation; scrubbing one entry through det = 0 collapses the parallelogram to a line, then flips orientation as det goes negative.
Kapittel 3 · Indreproduktrom
Best Approximation: The Perpendicular Wins
Best approximation theorem — the closest point in a subspace to a given vector is its orthogonal projection. A point u above a line L: dropping a perpendicular hits the foot uniquely, and any other candidate on the line is strictly farther by Pythagoras.
Gram-Schmidt: Subtract the Projection
Gram-Schmidt orthogonalisation — keep v₁ as u₁; subtract the projection of v₂ onto u₁ so what's left (u₂) is perpendicular; normalise to get an orthonormal pair. The same recipe extends to 3D by subtracting both shadows in turn.
Projection onto a Line
Project a vector u onto the line spanned by v. The foot of the perpendicular is proj_v(u) — the closest point. The leftover u minus its projection is orthogonal to v. proj_v(u) = (u·v / v·v) v.
Kapittel 4 · Differensialligninger
Diagonalisation: Eigenvectors as Natural Axes
A 2×2 matrix maps the unit circle to an ellipse. The two directions that stretch without rotating are the eigenvectors; their stretch factors are the eigenvalues. In the eigenbasis A is diagonal: A = PDP⁻¹.
Euler vs RK4: Same Step Size, Different Curves
Same ODE, same step size, two methods. Euler uses one tangent per step and drifts; classical Runge-Kutta samples four slopes per step, averages them, and hugs the true curve. A log-log error plot lays the order claim bare: slope 1 for Euler, slope 4 for RK4.
Euler's Method: Tangent Steps Along a Slope Field
Euler's explicit method — march along a slope field by taking tangent steps. Small h hugs the true curve; large h drifts off step by step. Global error is first order in h.
Phase Portraits of a 2×2 Linear System
For x' = A x with a 2×2 matrix A, the eigenvalues of A set the geometry of trajectories. Two negative real eigenvalues give a stable node; opposite signs give a saddle; a complex pair with negative real part gives a stable spiral.
Second-Order ODE as a First-Order System
A linear second-order ODE y'' + p y' + q y = 0 becomes a first-order system in two unknowns by introducing v = y'. The state vector (y, v) evolves under a single 2×2 matrix, opening up every first-order tool — Euler, RK4, eigenvector analysis.
Kapittel 5 · Funksjoner og derivasjon
Directional Derivative: A Tilted Slice
Directional derivative as the slope of a vertical slice. Pick a point on a surface z = f(x, y) and a unit direction u; the plane through that point parallel to u cuts the surface along a curve whose slope at the point is D_u f. Algebraically D_u f = ∇f · u, so every direction's slope is a projection of the gradient.
Gradient and Level Curves: Perpendicular by Design
On a contour plot, the gradient ∇f at every point is perpendicular to the level curve through that point. The gradient direction is steepest ascent; its length is the rate of steepest ascent. The directional derivative along the level-curve tangent is zero.
Linearisation: The Tangent Plane Snaps On
Linear approximation of a surface near a point. Lead with the one-variable picture — a curve and its tangent line — then zoom in to feel that the error vanishes faster than the step. Generalises to L(x, y) = f(a, b) + f_x(a, b)(x − a) + f_y(a, b)(y − b), the equation of the tangent plane.
Multivariable Limits Fail Along Different Paths
A two-variable limit must agree along every path. For f(x, y) = (xy + y³)/(x² + y²), approaching the origin along the x-axis gives 0, while approaching along y = x gives 1/2. Two different values — so the limit does not exist.
Kapittel 6 · Ekstremalpunkter
Global Extrema on a Triangle
Finding the global min and max of f(x, y) = x² + y² − xy + x + y on the triangle x ≤ 0, y ≤ 0, x + y ≥ −3. Walks Plenum 6 problem 3 — the highest-leverage exam workout in chapter 6. Candidates: an interior critical point at (−1, −1), three edge minimisers, three corners. Min = −1 at the interior point; max = 6 at the two off-origin corners.
Hesse Eigenvalues: Curvature on the Principal Axes
The Hesse matrix is symmetric, so it diagonalises in an orthonormal basis: its eigenvectors are the principal directions of curvature and its eigenvalues are the curvatures along them. Both eigenvalues positive — minimum; both negative — maximum; mixed signs — saddle. The discriminant just tracks the product λ₁ λ₂ = det H.
The Second-Derivative Test: A Discriminant Decides
Classifying a critical point with the Hesse matrix. Three cases shown side by side: a bowl (D > 0, f_xx > 0) is a local minimum; a dome (D > 0, f_xx < 0) is a local maximum; a saddle (D < 0) is neither. The discriminant D = f_xx · f_yy − f_xy² and the sign of f_xx read the verdict off the matrix.