Linear Algebra · Chapter 07

Vector spaces and bases

What a vector space is, beyond lists of numbers: subspaces, independence, bases, coordinates and dimension, and the rank seen as the size of what a matrix can reach.

span (v_{1}, \dots, v_{n})

dim (V)

v = i \sum c_{i} b_{i}

ker (A)

0 \in V

rank (A)

[v]_{B}

U \cap W

Lessons

Build the foundation

01Vector spacesArrows in the plane, lists of numbers, matrices of one shape, polynomials and functions can all be added and scaled, and in every case the same few rules hold. A vector space is any collection with an addition and a scaling that obey those rules, and whatever is proved from the rules then holds in all of them at once.Read lesson →
02SubspacesSome subsets of a space are worlds of their own: adding two of their vectors, or scaling one, never takes you outside. Those are the subsets worth a name.Read lesson →
03Linear independenceA list of vectors carries repeated information when one of them is already reachable from the others. Independence is the condition that none of them is, so every vector in the list earns its place.Read lesson →
04BasesA basis is a list that is large enough to reach every vector of a space and small enough that nothing in it is wasted. Those two demands pull in opposite directions, and a basis is where they meet.Read lesson →
05Coordinates in a basisBecause a basis describes each vector in exactly one way, the weights can be used as a name for it. Those weights are its coordinates, and they depend on which basis was chosen.Read lesson →
06Dimension

A space has many bases, and they may look nothing alike. What they always share is their size, and that shared number is what the word dimension means.

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07Column space and rankThe columns of a matrix span a subspace: everything the matrix can reach. Its dimension counts how many of the columns really pull in different directions, and it answers the question the lesson on systems opened, of which right-hand sides can be reached.Read lesson →

08Chapter practiceBring the chapter together: work in a vector space, test a subset, judge independence, recognise a basis, read coordinates, count a dimension and a rank. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Vector spaces and bases

Build the foundation