Row-Reduction Boot Camp
Constantin Teleman, Math 54.1, Fall 09
1. Notation and refreshers
Unless otherwise specified, Ais an m×nreal matrix, and vectors xare column vectors. Row vectors
are written as xT, where xis the corresponding column vector. The nullspace of A, Null(A)⊂Rn,
is the space of solutions of the homogeneous system Ax=0, the column space Col(A)⊂Rm
is the span of the columns. There are two more subspaces associated to A, this time consisting
of row vectors: the row space Row(A)⊂Rn, the span of the rows in A; and the left nullspace
LNull(A)⊂Rmthat we’ll meet below. (A quick and dirty definition is as the nullspace of the
‘flipped’ or transposed matrix AT: this is the matrix with switched indexing, (AT)ij =Aji .)
Caution! These four subspaces are distinct in general, with no obvious relation among them.
We will learn some subtle relations later.
The reduced row echelon form of Ais the matrix rref(A) produced from elementary row oper-
ations, with the properties that
•All zero-rows are at the bottom
•Every row which is not all zero starts with a 1, called the pivot or leading 1,
•Every pivot is strictly to the right of all pivots in the rows above it, and
•All entries above (and below) a pivot are zero.
The columns containing pivots are called pivot columns, the others are the free columns. The
nullspace of Aagrees with that of rref(A), and can be parametrized as follows: the free variables
can be chosen freely, and each equation in the reduced row system can then be used to solve for
the corresponding pivot variable.
A similar story applies to the inhomogeneous system Ax=b: it has the same general solution
as the reduced system rref(A)x=b0, where [rref(A)|b0] is the reduced form of the augmented
matrix [A|b]. The general solution can be parametrized by the same procedure.
We will learn a strong uniqueness property of rref (A): it is completely determined by nullspace
of A. (Similarly, the reduced form of the augmented matrix [A|b] is determined uniquely from the
affine space of solutions of Ax=b.)
A collection of vectors {v1,...,vr}is linearly independent if the only expression of 0as a linear
combination of the viis the one with 0 weights: that is,
kzv1+· · · +krvr=0⇒k1=k2=· · · =kr= 0.
A collection {w1,...,ws}spans the linear subspace L∈Rnif every wilies in Land every vector
in Lcan be expressed as a linear combination of the wi. A linearly independent, ordered collection
of vectors which spans Lis called a basis. Each vector in Lcan be uniquely expressed as a linear
combination of the basis elements (that is, the weights are uniquely determined). The main example
is the standard basis e1,...,enof Rn, the unit vectors on the coordinate axes.
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