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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**Example text**

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⎦.