New PDF release: A Practical Approach to Linear Algebra

By Prabhat Choudhary

ISBN-10: 8189473956

ISBN-13: 9788189473952

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Then each point in ]R3 is projected into a point on the plane such that the point, its image and the centre of the projection lie on the same line. This is exactly how a camera works, and it is a reasonable first approximation of how our eyes work. Let us get a formula for the projection. Assume that the focal point is (0, 0, d)T and that we are projecting onto x-y plane. "'" (x',y' , O) x' z x h) z Fig. Finding Coordinates x*, y* of the Perspective Projection of the Point (x, y, z) T xd x x*= - - = - - d-z l-z/d and similarly y* = y .

To get a more realistic picture one needs to use the so-called perspective projection. To: Qefine a perspective projection one needs to pick a point the centre of projection or the 24 Basic Notions focal point) and a plane to project onto. Then each point in ]R3 is projected into a point on the plane such that the point, its image and the centre of the projection lie on the same line. This is exactly how a camera works, and it is a reasonable first approximation of how our eyes work. Let us get a formula for the projection.

We therefore conclude that AB=[ ~ 41 53 2-Ij o -1 7 6 1 4 2 3 0 -2 3 +~ 17 Example. Consider again the matrices A~P -1 4 3 1 5 0 7 -Ij ~ and B= 1 4 2 3 0 -2 3 137 1. -12 56 Matries Note that B is a 4 x 2 matrix and A is a 3 x 4 matrix, so that we do not have a definition for the "product" BA. We leave the proofs of the following results as exercises for the interested reader. Proposition. (Associative Law) Suppose that A is an mn matrix, B is an np matrix and C is an p x r matrix. Then A(BC) = (AB)C.

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A Practical Approach to Linear Algebra by Prabhat Choudhary


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