l11 - Math 20F Linear Algebra Lecture 11: 4.3 Linearly...

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Math 20F Linear Algebra Lecture 11: 4.3 Linearly independent sets: Basis. 2.3 Characterizations of Invertible Matrices. Invertible Matrix Theorem (IMT) Let A be a given n × n matrix. Then the following are equivalent: a) A is invertible. b) A is row equivalent to I . c) A has n pivot positions d) The equation A x = 0 has only the trivial solution. e) The columns of A are linearly independent. f) The linear transformation x A x is one-to-one. g) The equations A x = b has a solution for each b . h) The columns of A span R n . i) The linear transformation x A x is onto. j) There is an n × n matrix C such that CA = I . k) There is an n × n matrix D such that AD = I . l) A T is invertible. Basis : Let H be a subspace of a vector space V . { b 1 , ··· , b n } is called a basis for H if (i) { b 1 , ··· , b n } are linearly independent, and (ii) { b 1 , ··· , b n } span H . Ex 3 Show that e 1 = 1 0 0 , e 2 = 0 1 0 , e 3 = 0 0 1 is a basis for
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l11 - Math 20F Linear Algebra Lecture 11: 4.3 Linearly...

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