Vector Fundamentals

Linear Algebra (Matrices) · Axiom Academy

A vector in is an ordered list of numbers — a column of components with a tip and a length. Here is what that means, exactly. A vector in is an ordered list of n real numbers, written as a column. Each entry v_i is a component ; their order matters, so and are different vectors. In and the components double as instructions: go v_1 along the x -axis, then v_2 along the y -axis, and the arrow from the origin to that endpoint is the vector. A vector in — a column of n components Equal componentwise — entry by entry, in order The magnitude (or length) of a vector, written , measures how long its arrow is. In the components v_1 and v_2 are the two legs of a right triangle, so the arrow is its hypotenuse — and the Pythagorean theorem hands you the length directly. The run v_1 along the x -axis — the first component. The rise v_2 along the y -axis — the second component. The arrow itself — its length is . Square every component, sum, take the root — the same rule in . For the legs are 3 and 2 , so . Notice always, and happens only for the zero vector . Collect every vector with n real components and you have the space . Each added component is one more axis, one more rung on the dimension ladder: is the plane, is the space around you, and carries on past what we can draw. We lose the picture beyond , but the definition — an ordered list of n reals — never changes. Two or three components draw a real arrow with a visible length and direction.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.