A basis is a set of vectors that spans a space and is independent. It reaches everything, with nothing to spare.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Both halves matter
Spanning without independence is wasteful. Independence without spanning is incomplete. A basis is exactly the balance point.
Dimension
Every basis of a given space has the same number of vectors. That number is the dimension.
The standard basis
⟨1, 0⟩ and ⟨0, 1⟩ form the standard basis of the plane. It is convenient, but far from the only choice.
Coordinates
Every vector is a unique combination of basis vectors. Those weights are its coordinates in that basis.
Why other bases matter
A well-chosen basis can make a hard problem obvious. Chapter 7 picks a basis of eigenvectors for exactly that reason.
The smallest set that still reaches everything
A basis is an independent set that spans the space. Independent means no redundancy and spanning means nothing is missed, so a basis is a description with no waste.
Bases are not unique
A space has infinitely many bases, and different ones suit different problems. Choosing a convenient basis is often the key step in simplifying a computation.
Dimension is well defined
Every basis of a given space has the same number of vectors. That number is the dimension, and the fact that it does not depend on the choice of basis is a genuine theorem.
A basis gives coordinates
Every vector has exactly one expression in terms of a basis, and those coefficients are its coordinates. Changing basis changes the coordinates while leaving the vector itself alone.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(1, 1)
- j-hat lands on(-1, 1)
- Determinant2
The shaded parallelogram is the image of the unit square, and its area is 2. That is exactly what the determinant measures.
Step 3: Watch an Example
One step at a time.
Watch Jae Test a Candidate Basis
Jae asks whether ⟨1, 1⟩ and ⟨2, 2⟩ form a basis of the plane.
- Step 1
He counts two vectors for a two-dimensional space, which is the right number.
Step 4: Your Turn
Practice makes it stick.
The Count
Problem 1 of 2
A basis of three-dimensional space. How many vectors does it contain?
The Line
Problem 2 of 2
A line through the origin in the plane. What is its dimension?
Find a Basis
1 of 8
⟨1, 0⟩ and ⟨0, 1⟩. Do they form a basis of the plane? 1 yes, 0 no.
2 of 8
⟨2, 4⟩ and ⟨1, 2⟩. Do they form a basis of the plane? 1 yes, 0 no.
3 of 8
Three vectors in the plane. Can they be a basis of it? 1 yes, 0 no.
4 of 8
One vector in the plane. Can it be a basis of the whole plane? 1 yes, 0 no.
5 of 8
The dimension of the whole plane. What is it?
6 of 8
A vector written as 3⟨1, 0⟩ + 5⟨0, 1⟩. What is its second coordinate in that basis?
7 of 8
Sort each set by whether it is a basis of the plane.
Tap something to move it.
- Empty
- Empty
8 of 8
The dimension of the space containing only the zero vector. What is it?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A basis of four-dimensional space. How many vectors does it contain?
Question 2 of 2
What two things must a basis do?
What You Learned
- A basis spans the space and is independent: it reaches everything with nothing to spare.
- Every basis of a space has the same size, and that number is the dimension.
- The weights on a basis are a vector’s coordinates in that basis.