Section 5.6 Rank and Nullity

FOUR FUNDAMENTAL MATRIX SPACES If we consider matrices A and A T together, then there are six vector spaces of interest: row space Arow space A T column space Acolumn space A T nullspace Anullspace A T. Since transposing converts row vectors to column vectors and vice versa, we really only have four vector spaces of interest: row space Acolumn space A nullspace Anullspace A T These are known as the fundamental matrix spaces associated with A.

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.

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).

RANK OF A MATRIX AND ITS TRANSPOSE Theorem 5.6.2: If A is any matrix, then rank(A) = rank(A T ).

DIMENSION THEOREM FOR MATRICES Theorem 5.6.3: If A is any matrix with n columns, then rank(A) + nullity(A) = n.

THEOREM Theorem 5.6.4: If A is an m×n 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.

MAXIMUM VALUE FOR RANK If A is an m×n matrix, then the row vectors lie in R n and the column vectors in R m. 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).

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.

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 m×1 matrix b. (b)The column vectors of A span R m. (c)rank(A) = m.

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.

THEOREM Theorem 5.6.8: If A is an m×n 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 m×1 matrix b.

THE “BIG” THEOREM Theorem 5.6.9: If A is an n×n matrix, and if T A : R n → R n 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 I n. (d)A is expressible as a product of elementary matrices. (e)Ax = b is consistent for every n×1 matrix b. (f) Ax = b has exactly one solution for every n×1 matrix b. (g)det(A) ≠ 0 (h)The range of T A is R n.

THE “BIG” THEOREM (CONCLUDED) (i)T A 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 R n. (m) The row vectors of A span R n. (n)The column vectors of A form a basis for R n. (o)The row vectors of A form a basis for R n. (p)A has rank n. (q)A has nullity 0.