Rank and Nullity

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Section 5.6

• Rank and Nullity

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FOUR FUNDAMENTAL MATRIX SPACES
If we consider matrices A and AT together, then
there are six vector spaces of interest row
space A row space AT column space A column
space AT nullspace A nullspace AT. Since
transposing converts row vectors to column
vectors and vice versa, we really only have four
vector spaces of interest row space A column
space A nullspace A nullspace AT These are
known as the fundamental matrix spaces associated
with A.
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ROW SPACE AND COLUMN SPACE HAVE EQUAL DIMENSION
Theorem 5.6.1 If A is any matrix, then the row
space and column space of A have the same
dimension.
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RANK AND NULLITY
The common dimension of the row space and column
space of a matrix A is called the rank of A and
is denoted by rank(A). The dimension of the
nullspace of A is called the nullity of A and is
denoted by nullity(A).
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RANK OF A MATRIX AND ITS TRANSPOSE
Theorem 5.6.2 If A is any matrix, then rank(A)
rank(AT).
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DIMENSION THEOREM FOR MATRICES
Theorem 5.6.3 If A is any matrix with n
columns, then rank(A) nullity(A) n.
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THEOREM
Theorem 5.6.4 If A is an mn matrix,
then (a) rank(A) the number of leading
variables in the solution of Ax
0. (b) nullity(A) the number of parameters in
the general solution of Ax  0.
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MAXIMUM VALUE FOR RANK
If A is an mn matrix, then the row vectors lie
in Rn and the column vectors in Rm. This means
the row space is at most n-dimensional and the
column space is at most m-dimensional.
Thus, rank(A) min(m, n).
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THE CONSISTENCY THEOREM
Theorem 5.6.5 If Ax b is a system of m
equations in n unknowns, then the following are
equivalent. (a) Ax b is consistent. (b) b is in
the column space of A. (c) The coefficient matrix
A and the augmented matrix A b have the
same rank.
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THEOREM
Theorem 5.6.6 If Ax b is a linear system of m
equations in n unknowns, then the following are
equivalent. (a) Ax b is consistent for every
m1 matrix b. (b) The column vectors of A span
Rm. (c) rank(A) m.
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PARAMETERS AND RANK
Theorem 5.6.7 If Ax b is a consistent linear
systems of m equations in n unknowns, and if A
has rank r, then the general solution of the
system contains n - r parameters.
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THEOREM
Theorem 5.6.8 If A is an mn matrix, then the
following are equivalent. (a) Ax 0 has only the
trivial solution. (b) The column vectors of A are
linearly independent. (c) Ax b has at most one
solution (none or one) for every m1 matrix b.
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THE BIG THEOREM
Theorem 5.6.9 If A is an nn matrix, and if TA
Rn ? Rn is multiplication by A, then the
following are equivalent. (a) A is
invertible (b) Ax 0 has only the trivial
solution. (c) The reduced row-echelon form of A
is In. (d) A is expressible as a product of
elementary matrices. (e) Ax b is consistent for
every n1 matrix b. (f) Ax b has exactly one
solution for every n1 matrix b. (g) det(A) ?
0 (h) The range of TA is Rn.
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THE BIG THEOREM (CONCLUDED)
(i) TA is one-to-one. (j) The column vectors of A
are linearly independent. (k) The row vectors of
A are linearly independent. (l) The column
vectors of A span Rn. (m) The row vectors of A
span Rn. (n) The column vectors of A form a basis
for Rn. (o) The row vectors of A form a basis for
Rn. (p) A has rank n. (q) A has nullity 0.