March 7, 2017

Algebraic projective geometry by the late J. G. Semple, G. T. Kneebone

By the late J. G. Semple, G. T. Kneebone

First released in 1952, this publication has confirmed a invaluable advent for generations of scholars. It presents a transparent and systematic improvement of projective geometry, development on ideas from linear algebra.

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Xn ⎡ ⎤ b1 ⎢ b2 ⎥ ⎢ ⎥ b = ⎢ . ⎥. ⎣ .. ⎦ bm Thus, if m = n, A will be rectangular and the dimensions of x and b will be different. Example 1 Convert the following system to matrix form: x + 2y − z + w = 4, x + 3y + 2z + 4w = 9. Solution A= Example 2 1 1 2 3 −1 2 1 , 4 ⎡ ⎤ x ⎢y⎥ ⎥ x=⎢ ⎣ z ⎦, w b= 4 . 9 Convert the following system to matrix form: x − 2y = −9, 4x + y = 9, 2x + y = 7, x − y = −1. Solution ⎡ 1 ⎢4 A=⎢ ⎣2 1 ⎤ −2 1⎥ ⎥, 1⎦ −1 ⎡ x= x , y ⎤ −9 ⎢ 9⎥ ⎥ b=⎢ ⎣ 7⎦. −1 A system of equations given by (1) or (2) can possess no solutions, exactly one solution, or more than one solution (note that by a solution to (2) we mean a vector x which satisfies the matrix equality (2)).

1 Solution We first partition A and B in the following manner ⎡ 3 1 ⎣ 1 4 A= 3 1 ⎤ ⎡ 2 1 3 −1⎦ and B = ⎣−1 0 2 0 1 ⎤ 2 1⎦; 1 then, ⎡ ⎢ ⎢ AB = ⎢ ⎣ ⎡ ⎢ =⎣ ⎡ 3 1 1 4 3 1 1 −1 1 −1 3 2 + 0 −1 0 3 + 2 0 0 1 2 −3 9 0 + 3 0 2 −1 2 9 + 0 2 2 11 = ⎣−3 2 2 11 ⎤ ⎡ 9 2 5⎦ = ⎣−3 2 9 1 7 2 + 6 −1 7 + 2 11 2 11 ⎤ 9 5⎦. 5 31 Submatrices and Partitioning Example 2 Find AB if ⎡ 3 ⎢2 ⎢ A =⎢ ⎢0 ⎣0 0 Solution ⎡ ⎢ ⎢ ⎢ ⎢ AB = ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ AB = ⎢ ⎢ ⎣ ⎡ 5 ⎢4 ⎢ =⎢ ⎢0 ⎣0 0 1 0 0 0 0 ⎤ 0 0⎥ ⎥ 3⎥ ⎥ 1⎦ 0 ⎡ 2 1 B = ⎣−1 1 0 1 and ⎤ 0 0⎦.

Find YD. 14. Find YC. 15. Find DX. 16. Find XD. 17. Find EF. 18. Find FE. 19. Find YF. 20. Find AB if A= 2 3 6 9 and B= 3 −1 −6 . 2 Note that AB = 0 but neither A nor B equals the zero matrix. 21. Find AB and CB if A= 3 1 2 , 0 2 1 4 , 2 2 4 x . y B= Thus show that AB = CB but A = C 22. Compute the product 1 3 C= 1 3 6 . 3 17 Matrix Multiplication 23. Compute the product ⎡ 1 ⎣3 1 0 1 3 ⎤⎡ ⎤ −1 x 1⎦ ⎣y⎦. 0 z 24. Compute the product a11 a21 a12 a22 x . y 25. Compute the product ⎡ b11 b21 b12 b22 ⎤ 2 ⎣−1⎦.

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