Fundamentals of Matrix Algebra 391
Linear Independence and Rank
Consider an n × m matrix A. A set of p columns extracted from the matrix A
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aaaaiini nipp(^111)
1
,,
,,
are said to be linearly independent if it is not possible to find p constants
β=s,1sp,..., such that the following n equations are simultaneously
satisfied:
ββ 11 aa,,ip 1 ... 1 ip 0
++ =ββ 1 aani,, 1 ++... pnip= 0Analogously, a set of q rows extracted from the matrix A are said to be
linearly independent if it is not possible to find q constants λ=s,1sq,...,
such that the following m equations are simultaneously satisfied:
λλ 11 aaiq 1 ,,... iq 1 0
++ =λλ 1 aaim 1 ,,++... qiqm= 0It can be demonstrated that in any matrix the number p of linearly inde-
pendent columns is the same as the number q of linearly independent rows.
This number is equal, in turn, to the rank r of the matrix. Recall that an
n × m matrix A is said to be of rank r if at least one of its (square) r-minors
is different from zero while all (r + 1)-minors, if any, are zero. The constant
p, is the same for rows and for columns. We can now give an alternative
definition of the rank of a matrix: Given a n × m matrix A, its rank, denoted
rank(A), is the number r of linearly independent rows or columns as the row
rank is always equal to the column rank.
Vector and Matrix Operations
Let’s now introduce the most common operations performed on vec-
tors and matrices. An operation is a mapping that operates on scalars,