Computer Science › Foundations of Computer Science
A vector as an ordered list of numbers, and the two operations — addition and scalar multiplication — every other idea in this discipline is built from.
Scaling and adding a set of vectors together (a linear combination), and the span — every vector reachable that way — which turns out to be exactly the set of solutions a system of equations can produce.
When no vector in a set is redundant (linear independence), the minimal spanning set that results (a basis), and the one number — dimension — that every basis of the same space agrees on.
Rewriting a whole system of linear equations as one matrix-vector equation Ax = b, and the two complementary ways to picture it — a row at a time, or a linear combination of columns.
A systematic algorithm for solving Ax = b by eliminating variables row by row — the mechanical procedure underneath every solver, from a hand calculation to a numerical library.
Formalizing what elimination was already doing implicitly — the precise rules for adding, scaling, and multiplying matrices, including why matrix multiplication isn't commutative.
Flipping a matrix across its diagonal (the transpose), and the special case — symmetric matrices, equal to their own transpose — that turns out to have unusually well-behaved eigenvalues later on.
The matrix that does nothing (the identity), and the matrix that undoes another one entirely (its inverse) — and precisely when that inverse fails to exist.
A single number computed from a square matrix that tells you, immediately, whether it has an inverse at all — and geometrically, the volume-scaling factor of the transformation it represents.
The set of every vector a matrix can produce (its column space), and the set of inputs it collapses to zero (its null space) — together, these say exactly which systems Ax = b can and cannot be solved.
The number of genuinely independent rows (equivalently, columns) a matrix has — a single number that predicts how many solutions Ax = b has, before any elimination is even attempted.
A matrix isn't just a grid of numbers — multiplying by it rotates, scales, or reflects space, which is exactly the mechanism behind every 2D and 3D graphics transformation.
The special directions a matrix's transformation only stretches or shrinks, never rotates away from (eigenvectors), and the factor by which it stretches them (eigenvalues).
A matrix whose columns are each a probability distribution, whose long-run behavior settles into a stationary distribution that is nothing more than its dominant eigenvector — the exact mechanism behind PageRank.
Rewriting a matrix in terms of its own eigenvectors and eigenvalues — a change of perspective that turns the otherwise painful task of computing a matrix's 100th power into simple arithmetic.
Multiplying two vectors into a single number (the dot product), and measuring a vector's own length (its norm) — the geometric building blocks behind angle, similarity, and distance.
The vector of steepest ascent for a function of several variables — a direct generalization of a single-variable derivative's slope, introduced here purely through vector notation, with no calculus prerequisite assumed.
Finding the best-fit line through data that has no exact solution, by projecting onto a column space and setting a gradient to zero — the capstone that ties projections and gradients into a single, genuinely useful algorithm.
Vectors at right angles to one another (orthogonality), and matrices built entirely from such vectors — whose inverse turns out to be nothing more than their own transpose.
Why a symmetric matrix's eigenvalues are always real and its eigenvectors always orthogonal — a special guarantee that underlies PCA and every optimization problem built on a symmetric matrix.
Part of the Computer Science curriculum.
A vector as an ordered list of numbers, and the two operations — addition and scalar multiplication — every other idea in this discipline is built from.
Scaling and adding a set of vectors together (a linear combination), and the span — every vector reachable that way — which turns out to be exactly the set of solutions a system of equations can produce.
When no vector in a set is redundant (linear independence), the minimal spanning set that results (a basis), and the one number — dimension — that every basis of the same space agrees on.
Rewriting a whole system of linear equations as one matrix-vector equation Ax = b, and the two complementary ways to picture it — a row at a time, or a linear combination of columns.
A systematic algorithm for solving Ax = b by eliminating variables row by row — the mechanical procedure underneath every solver, from a hand calculation to a numerical library.
Formalizing what elimination was already doing implicitly — the precise rules for adding, scaling, and multiplying matrices, including why matrix multiplication isn't commutative.
Flipping a matrix across its diagonal (the transpose), and the special case — symmetric matrices, equal to their own transpose — that turns out to have unusually well-behaved eigenvalues later on.
The matrix that does nothing (the identity), and the matrix that undoes another one entirely (its inverse) — and precisely when that inverse fails to exist.
A single number computed from a square matrix that tells you, immediately, whether it has an inverse at all — and geometrically, the volume-scaling factor of the transformation it represents.
The set of every vector a matrix can produce (its column space), and the set of inputs it collapses to zero (its null space) — together, these say exactly which systems Ax = b can and cannot be solved.
The number of genuinely independent rows (equivalently, columns) a matrix has — a single number that predicts how many solutions Ax = b has, before any elimination is even attempted.
A matrix isn't just a grid of numbers — multiplying by it rotates, scales, or reflects space, which is exactly the mechanism behind every 2D and 3D graphics transformation.
The special directions a matrix's transformation only stretches or shrinks, never rotates away from (eigenvectors), and the factor by which it stretches them (eigenvalues).
A matrix whose columns are each a probability distribution, whose long-run behavior settles into a stationary distribution that is nothing more than its dominant eigenvector — the exact mechanism behind PageRank.
Rewriting a matrix in terms of its own eigenvectors and eigenvalues — a change of perspective that turns the otherwise painful task of computing a matrix's 100th power into simple arithmetic.
Multiplying two vectors into a single number (the dot product), and measuring a vector's own length (its norm) — the geometric building blocks behind angle, similarity, and distance.
The vector of steepest ascent for a function of several variables — a direct generalization of a single-variable derivative's slope, introduced here purely through vector notation, with no calculus prerequisite assumed.
Finding the best-fit line through data that has no exact solution, by projecting onto a column space and setting a gradient to zero — the capstone that ties projections and gradients into a single, genuinely useful algorithm.
Vectors at right angles to one another (orthogonality), and matrices built entirely from such vectors — whose inverse turns out to be nothing more than their own transpose.
Why a symmetric matrix's eigenvalues are always real and its eigenvectors always orthogonal — a special guarantee that underlies PCA and every optimization problem built on a symmetric matrix.