6. Column Space and Nullspace

Overview

Gilbert Strang introduces vector spaces and subspaces, then explains the column space and nullspace of a matrix through a 4-by-3 example. He connects the column space to which right-hand sides make Ax=b solvable, shows that the example’s columns span a two-dimensional subspace of R^4, and identifies its nullspace as a line in R^3.

Key takeaways

A set of vectors is a subspace only if it contains zero and is closed under addition and scalar multiplication; a plane or line in R^3 must pass through the origin.
The intersection of any two subspaces is a subspace, but their union generally is not because combining vectors from different parts can leave the union.
For any matrix A, Ax=b is solvable exactly when b belongs to the column space of A, since Ax is a linear combination of A’s columns.
In the example with columns (1,2,3,4), (1,1,1,1), and (2,3,4,5), the third column is the sum of the first two, making the column space two-dimensional in R^4.
The nullspace consists of all coefficient vectors x satisfying Ax=0; in the example it is the line of vectors c(1,1,-1) in R^3.
Solutions to Ax=0 always form a subspace, whereas solutions to Ax=b for nonzero b generally form a shifted set that excludes zero.

Chapters

0:00 Vector Spaces and Subspaces Through the Origin

A vector space is closed under vector addition and multiplication by any scalar, so it contains every linear combination of its vectors.
In R^3, the whole space, any plane through the origin, and any line through the origin are subspaces.
Every subspace must contain the zero vector, since multiplying any vector by zero must stay inside the space.

4:00 Why Unions Fail but Intersections of Subspaces Work

The union of a plane P and a line L is generally not a subspace: adding a vector from each can produce a vector outside both.
For the illustrated plane and line, P intersect L contains only the zero vector, which is itself a subspace.
For any subspaces S and T, their intersection is closed under addition and scalar multiplication because vectors in the intersection belong to both spaces.

11:00 Defining the Column Space of a 4-by-3 Matrix

Strang uses the matrix with columns (1,2,3,4), (1,1,1,1), and (2,3,4,5), so its column space is a subspace of R^4.
The column space consists of all linear combinations of the matrix’s columns, not just the three columns themselves.
Because the matrix has only three columns in R^4, their combinations do not fill all of R^4.

16:00 The Column Space Gives the Solvability Test for Ax=b

For the 4-by-3 example, Ax=b is a system of four equations with three unknowns, so an arbitrary b in R^4 need not be reachable.
The right-hand side b is solvable exactly when it is a linear combination of the columns, equivalently when b belongs to the column space.
For b=(1,2,3,4), x=(1,0,0) is a solution because b is the first column; for b=(1,1,1,1), x=(0,1,0) works.

23:00 Dependent Columns and the Two-Dimensional Column Space

The third column (2,3,4,5) equals the sum of the first two columns, so it adds no new direction to the column space.
Strang calls the first independent columns pivot columns and keeps the first two under his convention.
The example’s column space is therefore a two-dimensional subspace of R^4.

28:00 The Nullspace Contains Solutions to Ax=0

The nullspace N(A) is the set of all vectors x satisfying Ax=0, so its elements are coefficient vectors rather than right-hand sides.
For an m-by-n matrix, the column space lies in R^m while the nullspace lies in R^n; here those spaces are in R^4 and R^3, respectively.
The zero vector is always in N(A), because multiplying any matrix by the zero vector gives zero.

32:00 Finding the Example Nullspace as a Line in R^3

For the 4-by-3 matrix, the coefficients (1,1,-1) give a zero combination: column 1 plus column 2 minus column 3 equals zero.
All nullspace vectors have the form (c,c,-c), or c(1,1,-1), for any scalar c.
Those vectors form a line through the origin in R^3, including zero when c=0.

37:30 Why Every Nullspace Is a Subspace

If Av=0 and Aw=0, then A(v+w)=Av+Aw=0, so the sum of two nullspace vectors remains in N(A).
If Av=0, then A(12v)=12Av=0; the same scalar-multiplication argument works for any scalar.
The distributive and scalar laws for matrix multiplication establish closure, so solutions to Ax=0 always form a subspace.

43:00 Nonzero Right-Hand Sides Produce Shifted Solution Sets

For b=(1,2,3,4), both (1,0,0) and (0,-1,1) solve Ax=b, but the solution set is not a subspace because it does not contain the zero vector.
The solutions to Ax=b can form a shifted line or plane that does not pass through the origin, unlike the nullspace solutions to Ax=0.
Strang presents two ways to specify subspaces: generate them from vectors using linear combinations, or define them as solutions satisfying homogeneous equations.

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.