Week_4 - Week 4 2.4 Elementary Matrices An elementary...

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Unformatted text preview: Week 4 2.4 Elementary Matrices An elementary matrix is an n × n matrix that can be obtained from the n × n identity matrix by performing a single elementary row operation. Example 1 1- 3 and 1 0 2 0 1 0 0 0 1 are elementary matrices. Fact: Performing an elementary row operation on an m × n matrix A is equivalent to multiplying A on the left by the corresponding elementary matrix. Example 2 Multiply A = 1 2 3 2- 1 3 6 1 4 4 0 on the left by the elementary matrix corresponding to the ERO R 3 + 3 R 1 . Note: Every elementary matrix E is invertible and the inverse of E is simply the elementary matrix that transforms E back to I . 1 Invertible Matrix Theorem Let A be an n × n matrix. Then the following are equivalent: (a) A is invertible (b) AX = 0 has just the trivial solution (c) The Reduced Row Echelon Form of A is I n Proof: A Method for Computing A- 1 Looking at the equation E k ··· E 2 E 1 A = I, we multiply both sides on the right by A- 1 to obtain E k ··· E 2 E 1 I = A- 1 This tells us that the same sequence of EROs that reduce A to I will reduce I to A- 1 . We must keep track of the EROs that reduced A to I and apply these same EROs in the same order to I to give us A- 1 . We can apply these EROs simultaneously to a double matrix [ A | I ] to produce [ I | A- 1 ]. If A does not reduce to I , we must have a row of 0s and so A is not invertible. 2 Example 3 Find the inverse of (a) A = 2 7 1 1 4- 1 1 3 (b) B = 1 6 4 2 4- 1- 1 2 5 2.5 Matrix Transformations A transformation is a function involving vectors instead of scalars. Recall: A function is a rule that assigns each element in a set A a unique element in a set B or f : A → B . A is called the domain of f and B is called the codomain . If f ( a ) = b , we say b is the image of a under f . Just like we extended a single equation to a system of equations, we will extend functions of the form f : R → R to transformations T : R n → R m . Single System Matrix Form Equation 2 x + 3 = 1 3 x 1- x 2 = 7 x 1 + 4 x 2 = 3 3- 1 1 4 x 1 x 2 = 7 3 Transformation y = f ( x ) = 2 x + 3 w 1 = 3 x 1- x 2 w 2 = x 1 + 4 x 2 w 1 w 2 = 3- 1 1 4 x 1 x 2 We can express a transformation in matrix form as T ( ~x ) = A~x , where A is the standard matrix of the transformation T . As long as we can express each component of the output vector in terms of all components of the input vector, we can come up with a standard matrix. Example 4 T x 1 x 2 x 3 = x 1 + 2 x 2 + x 3 x 1 + 5 x 2 is a transformation from to . (a) Write as a system of equations. (b) Write in matrix form and identify the standard matrix A . (c) Find the image of the vector ~x = 1 2 3 under T ....
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This note was uploaded on 08/05/2008 for the course MATH 115 taught by Professor Dunbar during the Fall '07 term at Waterloo.

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Week_4 - Week 4 2.4 Elementary Matrices An elementary...

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