Jump to content

Linear mapping/Matrix to basis/Several properties/Fact/Proof

From Wikiversity
Proof

Let and denote the bases of and respectively, and let denote the column vectors of . (1). The mapping has the property

where is the -th entry of the -th column vector. Therefore,

This is if and only if for all , and this is equivalent with

For this vector equation, there exists a nontrivial tuple , if and only if the columns are linearly dependent, and this holds if and only if is not injective.
(2). See exercise.
(3). Let . The first equivalence follows from (1) and (2). If is bijective, then there exists a (linear) inverse mapping with

Let denote the matrix for , and the matrix for . The matrix for the identity is the identity matrix. Because of fact, we have

and therefore is invertible. The reverse implication is proved similarly.