10. The Four Fundamental Subspaces

Overview

The lecture corrects a mistaken claim about three vectors in R^3, then develops the four fundamental subspaces of an m by n matrix A: its column space, null space, row space, and left null space. Using row reduction, it connects their dimensions to rank r, gives methods for finding bases, and closes by treating 3 by 3 matrices as vectors in a vector space.

Key takeaways

For an m by n matrix of rank r, the four fundamental subspaces are C(A) and N(Aᵀ) in R^m, plus C(Aᵀ) and N(A) in R^n.
The row space and column space both have dimension r; pivot columns of the original A give a column-space basis, while nonzero rows of its reduced form give a row-space basis.
Special solutions to Ax = 0, one for each free variable, form a basis for N(A), so its dimension is n-r.
The left null space N(Aᵀ) consists of vectors y satisfying yᵀA = 0, and has dimension m-r.
Appending the identity to A and applying the same row operations yields EA = R; rows of E corresponding to zero rows of R encode dependencies among A's rows.
The space of all 3 by 3 matrices has subspaces including upper triangular, symmetric, and diagonal matrices; the diagonal subspace has dimension 3.

Chapters

0:00 Correcting the R^3 Example with Dependent Rows

The proposed vectors (1,1,2), (2,2,5), and (3,3,8) are not independent, correcting the previous lecture's mistaken example.
The matrix formed from those vectors as columns has two identical rows, so its rank is at most 2 and its three columns cannot be independent.
The error illustrates how row dependence can reveal column dependence through the equality of row rank and column rank.

4:21 Defining the Four Fundamental Subspaces and Their Ambient Spaces

The four spaces are the column space C(A), null space N(A), row space C(Aᵀ), and left null space N(Aᵀ).
For an m by n matrix A, C(A) and N(Aᵀ) lie in R^m, while N(A) and C(Aᵀ) lie in R^n.
The row space consists of combinations of A's rows; writing it as C(Aᵀ) lets the rows be handled as column vectors.

8:28 Column-Space Bases and the Shared Rank

The column space has dimension r, the rank of A, and a basis comes from the pivot columns of the original matrix A.
The row space also has dimension r, so row and column rank are equal.
In the corrected R^3 example, two independent rows imply rank 2, which rules out three independent columns.

12:28 Null-Space Bases and the Rank-Nullity Count

Row-reducing A to reduced row echelon form reveals pivot and free variables for solving Ax = 0.
Setting each free variable to 1 in turn, with the other free variables set to 0, produces special solutions that form a basis for N(A).
There are n-r free variables, so dim N(A) = n-r; together with the row-space dimension r, the total is n.

21:09 Finding a Row-Space Basis from the Reduced Matrix

Elementary row operations change the column space but preserve the row space, because each new row is a combination of the original rows.
For the example matrix, reduction produces pivot rows (1,0,1,1) and (0,1,1,0), which form a basis for the row space.
The nonzero rows of the reduced matrix provide r independent spanning vectors; reversing the row operations shows the original rows lie in the same row space.

29:48 Why N(Aᵀ) Is Called the Left Null Space

A vector y belongs to the left null space when Aᵀy = 0.
Transposing that equation gives yᵀA = 0, showing that y acts as a row vector multiplying A from the left.
The left null space lies in R^m and has dimension m-r.

33:26 Tracking Row Operations to Find a Left-Null-Space Basis

Appending the m by m identity matrix to A and applying the same row operations gives a matrix E such that EA = R.
For the 3-row example of rank 2, the last row of E records the dependent-row combination and supplies a basis vector for the one-dimensional left null space.
Keeping track of E finds left-null-space vectors without separately row-reducing Aᵀ.

42:10 How the Four Subspace Dimensions Pair Across Rⁿ and Rᵐ

In R^n, the row space has dimension r and the null space has dimension n-r, so their dimensions sum to n.
In R^m, the column space has dimension r and the left null space has dimension m-r, so their dimensions sum to m.
The paired dimensions follow from applying pivot-variable and free-variable counting to A and Aᵀ.

43:14 Three-by-Three Matrices as Vectors and Subspaces

The set of all 3 by 3 matrices forms a vector space under matrix addition and scalar multiplication, without needing matrix multiplication.
Upper triangular matrices and symmetric matrices are subspaces of this matrix space; their intersection consists of diagonal matrices.
The diagonal 3 by 3 matrices have dimension 3, with a basis given by matrices having a single 1 in each respective diagonal position.

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