Page 297 - 35Linear Algebra

16.3 Summary 297

Reading homework: problem 2

Example 152 (Row rank equals column rank)
Suppose M is an m × n matrix. The matrix M itself is a linear transformation
n m
M : R → R but it must also be the matrix of some linear transformation
linear
L : V −→ W .
Here we only know that dim V = n and dim W = m. The rank of the map L is
the dimension of its image and also the number of linearly independent columns of
M. Hence, this is sometimes called the column rank of M. The dimension formula
predicts the dimension of the kernel, i.e. the nullity: null L = dimV −rankL = n−r.
To compute the kernel we would study the linear system

Mx = 0 ,

which gives m equations for the n-vector x. The row rank of a matrix is the number
of linearly independent rows (viewed as vectors). Each linearly independent row of M
gives an independent equation satisfied by the n-vector x. Every independent equation
on x reduces the size of the kernel by one, so if the row rank is s, then null L+s = n.
Thus we have two equations:

null L + s = n and null L = n − r .

From these we conclude the r = s. In other words, the row rank of M equals its
column rank.

16.3 Summary

We have seen that a linear transformation has an inverse if and only if it is
bijective (i.e., one-to-one and onto). We also know that linear transforma-
tions can be represented by matrices, and we have seen many ways to tell
whether a matrix is invertible. Here is a list of them:

Theorem 16.3.1 (Invertibility). Let V be an n-dimensional vector space and
suppose L : V → V is a linear transformation with matrix M in some basis.
Then M is an n × n matrix, and the following statements are equivalent:

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