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Download PDF by Prabhat Choudhary: A Practical Approach to Linear Algebra

By Prabhat Choudhary

ISBN-10: 8189473956

ISBN-13: 9788189473952

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Another often used transformation is reflection: for example the matrix defines the reflection through x-axis. We will show later in the book, that any linear transformation in ]R2 can be represented either as a composition of scaling rotations and reflections. However it is sometimes convenient to consider some di erent transformations, like the shear transformation, given by the matrix This transformation makes all objects slanted, the horizontal lines remain horizontal, but vertical lines go to the slanted lines at the angle j to the horizontal ones.

En} there. , it is just the matrix B whose kth column is the vector (column) vk . And in the other direction [l]BS = ([1] SB )-1 = 15 1. For example, consider a basis un B={(1), in }R2 , and let S denote the standard basis there. Then [1]SB = Ui) =: B and 1]-1 S[1]BS= [ SB 1(-I2 -I2) = B-1 ="3 (we know how to compute inverses, and it is also easy to check that the above matrix is indeed the inverse of B) An example: going through the standard basis. In the space of polynomials of degree at most 1 we have bases .

Then A(BC) = (AB)C. Proposition (Distributive Laws) (a) Suppose that A is an m x n matrix and Band Care n x p matrices. Then A(B + C) = AB + AC. (b) Suppose that A and Bare m x n matrices and C is an n x p matrix. Then (A + B)C = AC +BC. Proposition. Suppose that A is an m x n matrix, B is an n x p matrix, and that c E JR. Then c(AB) = (cA)B = A(cB). Systems of Linear Equations Note that the system of linear equations can be written in matrix form as Ax = b, where the matrices A, x and b are given.

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A Practical Approach to Linear Algebra by Prabhat Choudhary

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