Math416_MatrixProperties - Math 416 Abstract Linear Algebra...

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Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E1 Some properties of matrices Let A be an m × n matrix. We discuss two interesting properties that A can have, which will be stated in several equivalent ways. Property 1 Proposition: The following are equivalent. 1. The columns of A are linearly independent. 2. The system Ax = 0 has the unique solution x = 0. 3. The null space of A is trivial: Null A = { } ⊂ R n . 4. The system Ax = b has at most one solution for any b ∈ R m . 5. The echelon form of A has a pivot in every column . 6. A has full column rank, i.e. dim Col A = n . 7. A : R n → R m is injective 1 . 8. A is left invertible. Property 2 Proposition: The following are equivalent. 1. The columns of A span R m . 2. The system Ax = b has a solution for any b ∈ R m . 3. Col A = R m . 4. The echelon form of A has a pivot in every row . 5. A has full row rank, i.e. dim Row A = m ....
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This note was uploaded on 12/16/2011 for the course MATH 416 taught by Professor Frankland during the Fall '11 term at University of Illinois at Urbana–Champaign.

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Math416_MatrixProperties - Math 416 Abstract Linear Algebra...

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