4.8 rank, nullity, and the fundamental matrix spaces nullity of a and is denoted by nullity(a). rank

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• 4.8 Rank, Nullity, and

the Fundamental Matrix

Spaces

• Row and Column Spaces Have Equal Dimensions

THEOREM:

he row space and the column space of a matrix A have the same

dimension.

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• Definition: 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 null space of A is called the

nullity of A and is denoted by nullity(A).

Rank and Nullity

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• EXAMPLE 1 Find the rank and nullity of the matrix

Solution The reduced row echelon form of A is

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• THEOREM: Dimension Theorem for Matrices

If A is a matrix with π columns, then rank(A) + nullity(A) = π

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• EXAMPLE 3

The matrix

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• THEOREM:

If A is an π Γ π matrix, then

(a) rank(A) = the number of leading variables in the general

solution of π΄π = π.

(b) nullity(A) = the number of parameters in the general solution

of π΄π = π.

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• EXAMPLE 4

(a) Find the number of parameters in the general solution of π΄π = π if A is a 5 Γ 7 matrix of rank 3.

(b) Find the rank of a 5 Γ 7 matrix A for which π΄π = π has a two-dimensional solution space.

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• THEOREM:

If π΄π = π is a consistent linear system of π equations in π unknowns, and if A has rank r, then the general solution of the

system contains π β π parameters.

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• The Fundamental Spaces of a Matrix

THEOREM:

If A is any matrix, then ππππ(π΄) = ππππ(π΄π).

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• THEOREM:

If A is an π Γ π matrix, then: (a) The null space of A and the row space of A are orthogonal complements in πΉπ. (b) The null space of π΄π and the column space of A are orthogonal complements in πΉπ.

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• THEOREM: Equivalent Statements If A is an π Γ π matrix, then the following statements are equivalent. (a) A is invertible. (b π΄π₯ = 0 has only the trivial solution. (c) The reduced row echelon form of A is πΌπ. (d) A is expressible as a product of elementary matrices. (e) π΄π₯ = π is consistent for every n Γ 1 matrix b. ( f ) π΄π₯ = π has exactly one solution for every n Γ 1 matrix b. (g) det(A) = 0. (h) The column vectors of A are linearly independent. (i ) The row vectors of A are linearly independent. ( j) The column vectors of A span πΉπ. (k) The row vectors of A span πΉπ. (l ) The column vectors of A form a basis for πΉπ. (m) The row vectors of A form a basis for πΉπ. (n) A has rank n. (o) A has nullity 0. ( p) The orthogonal complement of the null space of A is πΉπ. (q) The orthogonal complement of the row space of A is {0}.

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• Applications of Rank

THEOREM:

Let A be an m Γ n matrix.

(a) (Overdetermined Case). If π > π , then the linear system π΄π = π is inconsistent for at least one vector b in πΉπ. (b) (Underdetermined Case). If π < π, then for each vector b in πΉπ

the linear system π΄π = π is either inconsistent or has infinitely many solutions.

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• EXAMPLE 7

((a) What can you say about the solutions of an overdetermined system π΄π = π of 7 equations in 5 unknowns in which A has rank r = 4?

(b) What can you say about the solutions of an underdetermined system π΄π = π of 5 equations in 7 unknowns in which A has rank r = 4?

Solution

(a) The system is consistent for some vector b in π7, and for any such b the number of parameters in the general solution is n β r = 5 β 4 = 1.

(b) The system may be consistent or inconsistent, but if it is consistent for the

vector b in π5, then the general solution has n β r = 7 β 4 = 3 parameters.

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