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Download Math 200 Spring 2010 March 12 Definition. An n by n matrix E is
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Math 200 Spring 2010 March 12 Definition. An n by n matrix E is called an elementary matrix if it is just one elementary row operation away from In . 1 0 0 1 0 0 Examples. The matrices E1 = 0 4 0 , E2 = −2 1 0 , and 0 0 1 0 0 1 0 1 0 E3 = 1 0 0 are elementary matrices. 0 0 1 a b c Effect of multiplication by an elementary matrix. For M = d e f g h i a b c a b c notice that E1 M = 4d 4e 4f , E2 M = −2a + d −2b + e −2c + f g h i g h i d e f and E3 M = a b c . g h i In general, if an elementary matrix E is obtained from In by a particular row operation, then multiplying any n-row matrix M on the left by E has the same effect as performing that row operation directly on M . Definitions. The span of a set of vectors is the set of all linear combinations of them. The row space of a matrix is the span of its row vectors, and the column space is the span of its column vectors. So the column space of a matrix is the same as its image (why?). Notation. We use the notation rref (A) to signify the reduced row echelon form of the matrix A. 1 2 −2 10 1 0 0 2 For example, If A = 3 7 −7 34 , then rref (A) = 0 1 −1 4 . 1 4 −4 18 0 0 0 0 What we can tell from rref (A): Let A be an m by n matrix. • The number of nonzero rows in rref (A) is the dimension of the row space of A. Those rows form a basis for the row space of A, which is a subspace of Rn . • The number of nonzero rows in rref (A) is also the dimension of the column space of A, which is a subspace of Rm . To find a basis for the column space of A, find which columns in rref (A) have leading ones, and then choose the corresponding column vectors from the original matrix A. , , So for the matrix A above, the row space of A is a 2-dimensional subspace 4 of R , with basis { 1 0 0 2 , 0 1 −1 4 }. The column space of 1 2 3 A is a 2-dimensional subspace of R , with basis 3 , 7 . 1 4 HOMEWORK DUE MONDAY, MARCH 15: 1. For each of the elementary matrices in the first examples above, find the inverse. Is the inverse an elementary matrix? If so, what is its defining elementary row operation? How does that relate to the defining row operation of the original matrix? 2. For each matrix A, find (i) a basis for ker(A), (ii) a basis for the image of A, and (iii) a basis for the column space of A. 1 −2 a) A = −2 4 2 4 −1 b) A = 5 0 3 1 12 −6 1 0 1 4 c) A = 4 1 1 3 3 2 0 4 3. If A is an m by n matrix, and rref (A) has k leading ones (meaning k nonzero rows), what is the dimension of a) ker(A)? b) the image of A? c) the column space of A? 4. Suppose that M is an n by n matrix that is row-equivalent to In . a) Is M invertible? Explain. b) What is ker(M )? c) What is the image of M ? d) Does M~x = ~b have a solution for every ~b ∈ Rn ? e) What is the row space of M ? 5. Suppose that C is a 3 by 5 matrix. Fill in the blanks: a) Multiplication on the left by C defines a linear transformation from R to R . b) The dimension of the kernel of C is at least and at most . c) The dimension of the image of C is at least and at most .