Every engineer's toolkit

A vector is a movement and a list of numbers, built from basic directions

Split it, solve the parts, add them back — same answer; the basis of everything

A matrix isn't a table; it moves all of space at once

The determinant is how much a transformation scales area

Solving a system is rewinding the transformation to find the input

The directions a transformation only stretches, never turns (λ = the stretch)

The same vector gets different coordinates depending on the basis that reads it; the right basis makes a transformation simple

A projection is the nearest point on a line or plane and the leftover is always perpendicular; stack it over data and you get least squares

Any matrix is three steps, rotate then stretch along axes then rotate, and the unit circle becomes an ellipse whose radii are the singular values

Find the directions where data spreads most; the principal components are the eigenvectors of the covariance matrix, and keeping them reduces dimension

The dot product bundles length and alignment at once; strip the lengths and you get cosine similarity, the core of search and recommendation

The inner product is the general structure that defines length, angle, and orthogonality; change the weights and the geometry bends, and even functions become vectors

Build an orthonormal basis from a skewed one by subtracting off projections; this is exactly the Q of QR

A = QR splits a matrix into an orthonormal Q and an upper-triangular R, solving least squares stably

Beyond 2x2 area to signed n-dimensional volume; compute by cofactor expansion and solve systems with Cramer rule

Scalars, vectors, and matrices are the first rungs of the tensor ladder; the outer product makes rank-1 and a matrix is their sum

A = PDP⁻¹ makes a transformation pure stretching in the eigenbasis; a symmetric matrix diagonalizes with perpendicular eigenvectors

It splits into column, null, and row spaces with rank + nullity = n, deciding whether a system has zero, one, or infinitely many solutions

xᵀAx is a bowl whose axes are eigenvectors and whose curvature is the eigenvalues; their signs decide minimum, maximum, or saddle

A complex number is a point on a plane; multiplying adds angles and multiplies lengths

e^{iθ} = cos θ + i sin θ, the language of AC through phasors and damped spirals

A function bends the grid yet preserves angles (conformal); carry a hard problem to an easy shape

Each point sets a slope and the solution flows along the field; the initial condition picks the curve

A spring and an RLC are the same equation; where the roots sit decides damping and oscillation

When the drive frequency matches the natural frequency the amplitude explodes; damping sets the peak (Q)

A coupled ODE is a phase-plane flow; Jacobian eigenvalues decide the fixed-point type and stability

A scalar field is a number per point, a vector field an arrow per point; a field is a function and a flow

The direction the value rises fastest; perpendicular to contours; −∇f is descent; ∇f=0 is the optimum

The net outflow at a point; a source is positive, a sink negative; the flux of a tiny loop is the divergence

The strength and sense of the local swirl; counter-clockwise is positive, clockwise negative; the circulation of a tiny loop is the curl

The work a vector accumulates along a path; path-dependent in general, but path-independent in a conservative field

The divergence summed inside equals the flux on the boundary, the curl inside equals the circulation; Gauss and Stokes tie inside to boundary

How far a point sits above or below its neighbors; the divergence of the gradient, behind diffusion and harmonic functions

A curl-free field is the gradient of a potential, is path-independent, and its closed-loop integral is zero

The curl of a gradient and the divergence of a curl are always zero; any field splits into a source part and a swirl part

Flux through a surface; the divergence theorem sums divergence over a volume, and Stokes ties boundary circulation to curl over a surface

The sample space is a board of area 1; an event is a region and probability is its area

Discrete is a PMF of bars; continuous has area under the curve as probability; histograms converge to a density; the CDF accumulates

E[X] is the balance point of probability weights; variance is the mean squared distance; SD is the width in the same units; aX+b transforms

The 68-95-99.7 rule; sums and sample means converge to normal (CLT); z-standardization; confidence intervals

Stack a polynomial around a point to mimic the curve; error grows as you leave the center

A power series gathers to one value inside its radius and diverges outside; the ratio test and the boundary

Jump where the tangent points to reach a root fast, with failure cases and basins of attraction

Send x through g(x) again and again and it converges to a fixed point when the slope is under 1

Step straight along a differential equation's slope; a big step makes the error explode

Rectangles, trapezoids and Simpson measure the area under a curve; more strips means more accuracy

Error shrinks by the method's order in the step size; accuracy versus stability versus cost

Fill the gaps with a function through the points; Lagrange bases, the Runge phenomenon, Chebyshev nodes and splines

The limits of a computer's floating point; machine epsilon, round-off accumulation, catastrophic cancellation and conditioning