\( \newcommand{\vect}[1]{\mathbf{#1}} \newcommand{\mat}[1]{\mathbf{#1}} \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \newcommand{\Prob}{\mathbb{P}} \newcommand{\T}{^{\top}} \newcommand{\norm}[1]{\lVert #1 \rVert} \newcommand{\abs}[1]{\lvert #1 \rvert} \newcommand{\ihat}{\mkern1mu\hat{\boldsymbol{\imath}}\mkern2mu} \newcommand{\jhat}{\mkern1mu\hat{\boldsymbol{\jmath}}\mkern2mu} \newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\argmax}{\operatorname*{arg\,max}} \newcommand{\argmin}{\operatorname*{arg\,min}} \newcommand{\cdata}[1]{\class{c-data}{#1}} \newcommand{\cmodel}[1]{\class{c-model}{#1}} \newcommand{\ctruth}[1]{\class{c-truth}{#1}} \newcommand{\cgrad}[1]{\class{c-grad}{#1}} \newcommand{\closs}[1]{\class{c-loss}{#1}} \newcommand{\cparam}[1]{\class{c-param}{#1}} \newcommand{\csecond}[1]{\class{c-second}{#1}} \renewcommand{\,}{\mkern3mu} \)

M02.2 · Linear combinations, span and basis

A free, illustrated module of 4 lessons and one interactive in M02 Linear Algebra I: Vectors and Matrices: Linear combinations; Span: everything you can reach; Linear independence; Basis and dimension.

M02.2 L1 · Core 4 lessons · about 100 min Live · in review

Part of M02 Linear Algebra I: Vectors and Matrices in the Mathematics track.

Scale some vectors and add them: that one move, the linear combination, is what all of linear algebra is built from. These four lessons ask what the move can reach (the span), when a vector adds nothing new (linear dependence), and how many vectors it takes to describe a space with nothing missing and nothing spare (a basis, and its size, the dimension).

What you need first. Vectors as arrows and as lists of numbers, and adding and scaling them tip to tail and in coordinates (M02.1). Solving two equations in two unknowns from school algebra.

Lessons

1. Linear combinations M02.2.1 25 min

   Scale some vectors and add the results: that single move is a linear combination, and the rest of linear algebra is built from it. Learn to compute one, to draw it tip to tail, and to find the weights that reach a target, by picture and by solving two equations.

2. Span: everything you can reach M02.2.2 25 min

   Fix a few vectors and let their weights run over all numbers. The set of points you can reach is their span. See why one vector spans a line, why two usually span a plane, what happens in three dimensions, and when adding a vector adds nothing at all.

3. Linear independence M02.2.3 25 min

   A set of vectors is linearly dependent when one of them is spare: it can be built from the others. Learn the two ways to say this, prove that they are the same statement, test small sets by hand and in NumPy, and see why spare columns make a model’s weights ambiguous.

4. Basis and dimension M02.2.4 25 min

   A basis is a set of vectors that reaches everything and has nothing spare. Prove that it gives every vector exactly one list of coordinates, meet the standard basis, and learn what dimension really counts: the number of vectors in any basis, not the number of entries.

The interactive

M02.2-X1 · Span explorer. Drag two vectors and move two weights. The shaded region is everything their combinations can reach: the whole plane, or a single line when the vectors are parallel. A target shows which weights reach it, or why none do. It lives in Span: everything you can reach.