Section 2.3: Characterizations of Invertible Matrices We Can Link

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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.

(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.

     1 3 0   −    Example: Determine if A =  4 11 1  is invertible. Explain.    −   2 7 3 

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− .