R05 - ENEE 241 02* READING ASSIGNMENT 5 Tue 02/17 Lecture...

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ENEE 241 02* READING ASSIGNMENT 5 Tue 02/17 Lecture Topic: matrix-vector product; matrix of a linear transformation; matrix-matrix product Textbook References: sections 2.1, 2.2.1, 2.2.2 Key Points: The matrix-vector product Ax , where A is a m × n matrix and x is a n -dimensional column vector, is computed by taking the dot product of each row of A with x . The result is a m -dimensional column vector. For a fixed matrix A , the product Ax is linear in x : A ( c 1 x (1) + c 2 x (2) ) = c 1 Ax (1) + c 2 Ax (2) In other words, A acts as a linear transformation, or linear system, which maps n -dimensional vectors to m -dimensional ones. Every linear transformation, or linear system, R n R m has a m × n matrix A associated with it. Each column of A is obtained by applying that transformation to the respective standard n -dimensional unit vector. If A is m × p and B is p × n , then the product AB is a m × n matrix whose ( i,j ) th element is the dot product of the i th row of A and the j th column of B . Theory and Examples: 1 . A m × n matrix consists of entries (or elements) a ij , where i and j are the row and column indices, respectively. The space of all real-valued m × n matrices is denoted by R m × n . 2 . A column vector is a matrix consisting of one column only; a row vector is a matrix consisting of one row only. The transpose operator · T converts row vectors to column vectors and vice versa. By default, a lower-case boldface letter such as a corresponds to a column vector. In situations where the orientation (row or column) of a vector is immaterial, we simply write
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R05 - ENEE 241 02* READING ASSIGNMENT 5 Tue 02/17 Lecture...

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