Section 5.6

1. Section 5.6 Rank and Nullity

2. 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.

3. 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.

4. 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).

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

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

7. 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.

8. MAXIMUM VALUE FOR RANK If A is an m×n 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).

9. 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.

10. 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 Rm. (c) rank(A) = m.

11. 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.

12. 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.

13. THE “BIG” THEOREM Theorem 5.6.9: If A is an n×n 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 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 TA is Rn.

14. 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.