Independence, basis and dimension: how many directions a space needs
Slide 4 kept counting: free columns, special solutions, pivots. Now we give that counting a name. Think of giving a friend walking directions: "go east, go north, and also go north-east". The third direction adds nothing, because north-east is just some east plus some north. Directions where none can be made from the others are independent. A set of independent directions that can still take you everywhere in a space (we say they span it) is a basis. The number of directions in a basis is the dimension: 2 for a flat map, 3 for the air a drone flies in. It is the same for every basis you pick (Strang §3.4 proves it).
Lesson 4 · slide 5Basis and dimension- Independent: the only combination giving 0 is the one with all coefficients (the numbers in the combination) 0.
- Span: all combinations of the vectors.
- Basis: independent vectors that span the space.
- Dimension: the number of vectors in any basis. The rank r is the dimension of C(A).
Dependent: one vector is a mix of the others
1−1 and −22 point along the same line: −22 = −2·1−1. So they are dependent, and there is a combination with nonzero numbers that gives 0:
Two ways to say independent
"No vector is a combination of the others" and "only the all-zero combination gives 0" mean the same thing. If v₃ = v₁ + v₂, then v₁ + v₂ − v₃ = 0: a combination with nonzero coefficients. The other way round: if a combination with a nonzero coefficient gives 0, solve it for that vector. It is a combination of the others.
The span of some vectors is all their combinations. 1−1 and −22 span a line. 11 and 1−1 span the whole plane.
Testing for independence
Put the vectors in the columns of A. These all say the same thing:
- The only solution of Ax = 0 is x = 0.
- N(A) contains only the zero vector.
- Every column has a pivot, so the rank is r = n, the number of vectors.
Example: the columns 12 and 34, so A = 1324. Elimination: row 2 − 2·row 1 gives 0−2. Two pivots, 1 and −2, so the columns are independent.
ℝⁿ means all vectors with n components. ℝ² is the plane and ℝ³ is 3D space.
Four vectors in ℝ³ are always dependent. Put them in a 3 by 4 matrix. Each pivot needs its own row, so there are at most 3 pivots for 4 columns. At least one column is free, and its special solution is a nonzero x with Ax = 0. In the same way, three vectors in the plane are always dependent.
Dependent in 3D
A basis is not unique
10 and 01 are a basis of the plane. So are 11 and 1−1. Different vectors, but always two. In the second basis, every vector still has exactly one recipe. For 31:
Exactly one, because the vectors are independent: two different recipes for the same point would subtract to a combination that gives 0.
Keep reading with the full course
Finish lesson 4 and you will be able to: write down every solution of Ax = b, not just one.
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Examples
Example 1 · A basis of the plane
10 and 11.
Show solutionHide solution
A basis.
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