Linearly Independent Sets and Bases

Main idea

A basis is a set of vectors that can build every vector in the space, with nothing redundant: it spans, and it's linearly independent. That combination is exactly what gives every vector one and only one recipe.

{\htmlData{sym=0}{\mathcal B}} = \{\vec b_1, \dots, \vec b_n\}: \text{ independent and } \operatorname{Span}{\htmlData{sym=0}{\mathcal B}} = V

What the symbols mean

$\mathcal B$
Basis
A minimal spanning set, or equivalently a maximal independent set.

Explanation

A basis is simply a set of building blocks for a vector space: enough vectors to build every vector in it, with none to spare. Think of it like the base of a building, the part everything else rests on. It's useful because once you've picked a basis, every vector has exactly one recipe in terms of it, and next lesson turns those recipes into coordinates. All we're really saying in the formula is two conditions: "independent" (no vector in the set is a combination of the others, so nothing is redundant) and $\operatorname{Span}\mathcal B = V$ (combinations of them reach everything in $V$ ). Two simple checks, and you've seen both before; bear with it.

Two ways to fail: too many vectors (dependent) or too few (they don't span).

Spanning set theorem. If one vector in a spanning set is a combination of the others, you can remove it and still span. Think about why. Say $\vec v_3 = 2\vec v_1 + \vec v_2$ . Anywhere a recipe used $\vec v_3$ , you could swap in $2\vec v_1 + \vec v_2$ instead, so dropping $\vec v_3$ loses nothing. Keep removing redundant vectors until the set is independent, and what's left is a basis.

Standard bases: $\vec e_1, \dots, \vec e_n$ for $\mathbb R^n$ ; $1, t, t^2, \dots, t^n$ for $\mathbb P_n$ .

For $\operatorname{Col}A$ : the pivot columns of $A$ form a basis.

A common mistake. After row reducing, it's tempting to think the basis is the pivot columns of the reduced matrix, because that's the matrix sitting in front of you. It isn't: row operations change the column space. In the example below, every column of the reduced matrix ends in a $0$ or $1$ , while the original columns are $\binom12$ , $\binom24$ and $\binom37$ . What row reduction does keep is which columns depend on which, so it tells you which columns to pick. Then take those columns from the original matrix.

There's an animation further down that shows what a basis buys you: one recipe for every point.

See it

Here x travels around a circle of radius 2.5, always built from b₁ = (2, 1) and b₂ = (−1, 1.5). Notice the weights keep changing, but at every moment exactly one pair works: when x = (2.5, 0), for instance, it's 0.9375·b₁ − 0.625·b₂. That's both halves of being a basis: b₁ and b₂ span the plane (some recipe always exists) and they're independent (only one does). Add a third vector and recipes stop being unique; drop one and most points become unreachable.

Worked examples

Example 1

Find a basis for $\operatorname{Col}A$ , $A = \begin{bmatrix}1 & 2 & 3\\ 2 & 4 & 7\end{bmatrix}$ .

1
Row reduce to $\begin{bmatrix}1 & 2 & 3\\ 0 & 0 & 1\end{bmatrix}$ , which has pivots in columns 1 and 3.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Is $\{(1, 0), (0, 1), (1, 1)\}$ a basis for $\mathbb R^2$ ?

Problem 2+5 XP

Is $\{(1, 2), (3, 4)\}$ a basis for $\mathbb R^2$ ?

Problem 3+5 XP

How many vectors are in the standard basis of $\mathbb P_2$ ?

Problem 4+5 XP

Which columns of $\begin{bmatrix}1 & 2 & 3\\ 2 & 4 & 7\end{bmatrix}$ form a basis for its column space? Give their numbers.

Problem 5+5 XP

If a spanning set contains a vector that's a combination of the others, does removing it change the span?

Problem 6+5 XP

A paint shop has three base paints with pigment amounts (red, yellow, blue) $P_1 = (2, 1, 0)$ , $P_2 = (0, 1, 1)$ and $P_3 = (2, 3, k)$ . For which $k$ is $P_3$ redundant, a mix of $P_1$ and $P_2$ , so that the three don't form a basis?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

For which value of $h$ do $1 + t$ , $t + t^2$ and $h + t^2$ fail to be a basis of $\mathbb P_2$ ?

Where it's used

Choosing a good basis is half of applied math: Fourier bases for sound, wavelets for images, and eigenvector bases for vibrations make hard problems simple.

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