9. Independence, Basis, and Dimension

Overview

MIT OpenCourseWare’s linear algebra lecture connects linear independence, span, basis, and dimension to matrix elimination and null spaces. It establishes that a matrix with more columns than rows has a nonzero nullspace vector, defines independence through the equation A c = 0, and shows that a matrix’s rank is the dimension of its column space while its nullity is n - r.

Key takeaways

For an m-by-n matrix with n > m, at least n - m variables are free in A x = 0, guaranteeing a nonzero nullspace vector and therefore dependent columns.
A set of vectors is independent exactly when the only coefficient vector c satisfying A c = 0 is c = 0, where the vectors are the columns of A.
A basis is both independent and spanning; for R^n, n vectors form a basis exactly when their square column matrix is invertible.
Every basis of the same vector space has the same number of vectors, and that shared count defines the space’s dimension.
The rank of A is the dimension of its column space, while the nullspace dimension is n - r, the count of free variables.
Once a space’s dimension is known, any independent collection with exactly that many vectors is automatically a basis.

Chapters

0:00 More Unknowns Than Equations Guarantee a Nonzero Nullspace

For an m-by-n matrix with n > m, row reduction can produce at most m pivot variables, so at least n - m variables are free.
Setting a free variable to a nonzero value and solving for the pivot variables gives a nonzero solution to A x = 0.
This nullspace fact will connect matrix elimination to linear dependence.

4:30 Linear Independence Means Only the Zero Combination Gives Zero

Vectors v1 through vn are independent if c1v1 + ... + cnvn = 0 implies every coefficient c1 through cn is zero.
If some nonzero coefficient vector c produces a zero combination, the vectors are dependent.
The definition applies to a collection of vectors, not to a matrix itself.

7:00 Scalar Multiples and Zero Vectors Make Collections Dependent

The pair v and 2v is dependent because 2v - 2v = 0 gives a nontrivial combination.
Any collection containing the zero vector is dependent, since a nonzero coefficient on that vector and zero coefficients elsewhere gives zero.
Two nonparallel vectors in the plane can be independent, but any three vectors in the plane must be dependent.

11:30 Test Column Independence with the Nullspace of A

Place vectors v1 through vn into the columns of a matrix A; a relation among them is exactly a solution to A c = 0.
The columns are independent precisely when the nullspace of A contains only the zero vector.
Independent columns are all pivot columns, so rank r equals n; dependent columns leave free variables and give r < n.

18:00 Span Is the Set of All Linear Combinations

Vectors span a space when every vector in that space is a linear combination of them.
The column space of a matrix consists of all combinations of its columns, so those columns span it by definition.
The span is the smallest space containing the chosen vectors, because any such space must contain all their combinations.

22:00 A Basis Must Be Both Independent and Spanning

A basis is a collection of vectors with two properties: the vectors are independent and they span the space.
The standard coordinate vectors form a basis for R^3; their column matrix is the identity, whose nullspace is zero.
For R^3, three vectors form a basis when their 3-by-3 column matrix is invertible.

30:00 Different Bases for One Space Have the Same Size

Many different vector collections can be bases for R^3; any invertible 3-by-3 matrix supplies one through its columns.
Two independent vectors such as (1,1,2) and (2,2,5) span a plane and form a basis for that plane, but not for all of R^3.
Every basis for a given space has the same number of vectors, and that number is the space’s dimension.

37:00 Pivot Columns Give a Basis for the Column Space

In the example matrix, four columns span the column space, but the columns are dependent because some can be expressed using others.
The first two pivot columns form a basis for the column space, and the matrix has rank 2.
Rank is a property of the matrix, while the dimension belongs to a space: rank A equals the dimension of A’s column space.

43:00 Independent Vectors and the Right Count Form a Basis

A basis for the example’s two-dimensional column space can use vectors other than the original pivot columns.
Once a space’s dimension is known to be 2, any two independent vectors in that space automatically span it and form a basis.
Independence prevents redundancy, while the dimension tells how many vectors are needed.

47:00 Nullspace Dimension Equals the Number of Free Variables

Each free variable in A x = 0 supplies a special solution, and those special solutions form a basis for the nullspace.
The nullspace dimension is the number of free variables, which is the number of columns n minus the rank r.
The lecture’s example has two free variables, so its nullspace is two-dimensional and its special solutions give a two-vector basis.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, MIT OpenCourseWare.