Download A Practical Approach to Linear Algebra by Prabhat Choudhary PDF

By Prabhat Choudhary

ISBN-10: 8189473956

ISBN-13: 9788189473952

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Since an invertible matrix must be square, it follows from counting dimensions, that similar matrices A and B have to be square and of the same size. , if A = g-IBQ, then B = QAg-I = (g-ItIA(g-I) (since g-l is invertible), therefore B is similar to A. So, we can just say that A and B are similar. The above reasoning shows, that it does not matter where to put Q and where g-l: one can use the formula A = QBg-I in the definition of similarity. The above discussion shows, that one can treat similar matrices as different matrix representation of the same linear operator (transformation).

To perform such projection one just needs to replace z coordinate by 0, the matrix of this projection is [ ~ ~ ~J, 000 y :r % Fig. Perspective Projection onto x - y plane: F is the centre (focal point) of the projection Such method is often used in technical illustrations. Rotating an object and projecting it is equivalent to looking at it from di erent points. However, this method does not give a very realistic picture, because it does not take into account the perspective, the fact that the objects that are further away look smaller.

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. We shall establish the following important result. Proposition. Every system of linear equations of the form,has either no solution, one solution or infinitely many solutions. Proof Clearly the system (2) has either no solution, exactly one solution, or more than one solution.

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