Linear Algebra · Chapter 02

Vectors and linear systems

From a list of numbers to linear combinations, and from a system of equations to the question of whether it can be solved at all.

A x = b

u + v

c v

span (u, v)

(x y)

x_{1} a_{1} + x_{2} a_{2} = b

R_{2} \leftarrow R_{2} - c R_{1}

R^{n}

Lessons

Build the foundation

01Vectors and coordinatesA vector is a list of numbers written in a fixed order. Each position answers a different question, so the numbers cannot be rearranged without describing something else.Read lesson →
02Adding and scalingTwo vectors of the same length are added one position at a time, and a vector is scaled by multiplying every component by a single number. Both results stay in the same space.Read lesson →
03Linear combinations and spanScale a few fixed vectors, add the results, and you reach a new vector. Everything reachable that way from one fixed list is called its span, and it is usually far larger than the list.Read lesson →
04Systems of linear equationsA linear equation constrains several unknowns without ever multiplying two of them together. Several such equations, read at once, ask for the lists of numbers that satisfy all of them.Read lesson →
05EliminationA system can be rewritten as a simpler one with exactly the same solutions. Three moves are enough, and each of them can be undone, which is why nothing is lost and nothing is invented.Read lesson →
06Chapter practice

Bring the chapter together: compare vectors, compute combinations, decide what a span holds, and read a system both as equations and as vectors. No worked example sits above the answers.

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Mapping the lesson.

Vectors and linear systems

Build the foundation