C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes2-3 - S...

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SECTION 2.3 A SUMMARY OF EVERYTHING! THE INVERTIBLE MATRIX THEOREM. If the matrix A is n × n , then either all the following statements are true for A or all the following statements are false for A . 1. A is invertible. 2. A is row equivalent to I . 3. A has n pivot positions. 4. The equation A x = 0 has only the trivial solution. 5. The columns of A are LI. 6. The linear transformation x A x is one-to-one. 7. The equation A x = b has at least one solution for every b in R n . 8. The columns of A span R n . 9. The range of the linear transformation x A x is all of R n . 10. There is an n × n matrix C such that CA = I . 11. There is an n × n matrix D such that AD = I . 12. A T is invertible. As review of all this stuff, let’s discuss some of these equivalences.
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A linear transformation T : R n R n is a function so we can think about whether it has an inverse function. As a reminder, this means there is a function
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Unformatted text preview: S : R n → R n such that S ( T ( x )) = x and T ( S ( y )) = y for all x and y in R n . S is called the (function) inverse of T . Suppose the linear transformation T is given by T ( x ) = A x for some matrix A . Then T is invertible exactly when A is invertible, and in this case the function inverse of T is given by S ( y ) = A-1 y . EXAMPLES. Is 2 6 4 0 7 8-6 5 invertible? Do as little work as possible! A matrix is lower triangular if the entries above the main diagonal are all 0’s. When is a square lower triangular matrix invertible? Why? Show that if A and B are square and AB is invertible, then so is B . HOMEWORK: SECTION 2.3...
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This note was uploaded on 04/15/2008 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas at Austin.

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C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes2-3 - S...

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