Section 2.3: Characterizations of Invertible Matrices We can link together the following concepts for n n matrices A: × Solving Ax = b • Linear independence of the columns of A • Linear (or matrix) transformations T(x) = Ax • Invertibility of A • Onto transformation: A transformation T : Rn Rm is onto Rm ! if each b in Rm is the image of at least one x in Rn, i.e., if T(x) = b has at least one solution for each b in Rm, i.e., if the range of T = codomain of T = Rm. One-to-one transformation: A transformation T : Rn Rm is one-to-one ! if each b in Rm is the image of at most one x in Rn, i.e., if T(x) = b has either no solution or a unique solution. 1 Theorem 8: The Invertible Matrix Theorem Let A be an n n matrix. Then the following statements are equivalent × (i.e., for a given A, they are either all true or all false). (a) A is an invertible matrix. (b) A is row equivalent to In. (c) A has n pivot positions. (d) The equation Ax = 0 has only the trivial solution. (e) The columns of A form a linearly independent set. (f) The linear transformation T(x) = Ax is one-to-one. (g) The equation Ax = b has at least one solution for each b in Rn. (h) The columns of A span Rn. (i) The linear transformation T(x) = Ax maps Rn onto Rn. (j) There is an n n matrix C such that CA = In. × (k) There is an n n matrix D such that AD = In. × (l) AT is an invertible matrix. 2 Fact following from IMT: Let A and B be n n matrices. If AB = In, then A and B are both × 1 1 invertible, and A− = B and B− = A. 2 3 6 7 6 1 3 0 7 6 − 7 6 7 Example: Determine if A = 6 4 11 1 7 is invertible. Explain. 6 7 6 − 7 46 2 7 3 57 3 Example: Suppose that H is a 5 5 matrix, and suppose that there is × a vector v in R5 which is not a linear combination of the columns of H. What can you say about the number of solutions to Hx = 0? Explain. 4 Note about invertible matrices: Let A be any n n invertible matrix. Then for all x in Rn: × 1 1 A− Ax = x and AA− x = x. Note about linear transformations: Every linear transformation can be written as a matrix transformation. Theorem: Let T : Rn Rm be a linear transformation. ! Then, there exists a unique m n matrix A such that × T(x) = Ax for all x in Rn. In fact: A = [ T(e ) T(e ) T(en)] 1 2 ··· where ej, j = 1; 2; :::; n, is the j-th column of In. A is called the standard matrix for the linear transformation T. 5 Invertible linear transformations: A linear transformation T : Rn Rn is said to be invertible if there ! is a function S : Rn Rn such that ! S(T(x)) = x for all x in Rn and T(S(x)) = x for all x in Rn. Questions: When is T invertible? If T is invertible, what is S and is S unique? Theorem 9: Let T : Rn Rn be a linear transformation, and let A be the standard ! matrix for T. Then T is invertible if and only if A is an invertible matrix. 1 In that case, S(x) = A− x is the unique function satisfying the definition. 1 S is called the inverse of T and is denoted by T− . 6.