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Read e-book online A First Course in Linear Algebra PDF

By Daniel Zelinsky and Samuel S. Saslaw (Auth.)

ISBN-10: 1483229564

ISBN-13: 9781483229560

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Prove that for every two vectors u and v, ||u + v | | 2 + | | u - v | | 2 = 2(||u|p+||v||2). 4. 19. 18 to show that the sum of the squares of the lengths of the four sides of any parallelogram equals the sum of the squares of the lengths of the diagonals. 7 . 1 DEFINITION u X V = (0263 — A362)i + (azbi — aibz)j + {αφι — αφι)}α where the a's are the components of u and the Vs are the com­ ponents of V. Admittedly this arrives at a reasonably important product through the back door. It would have been more in the spirit of our previous definitions if we had defined the cross product by specifying its geometrical properties, independent of compo­ nents or coordinate systems.

12 ( u X v ) · w = ± volume of parallelepiped with three edges u , V, w. 1. Compute i X (i + j + k) (2i ~ j + k ) X (3i + 3j [2, 0, 1 ] X [ 1 , 1, 2 ] k X (aii + a2J + azk). 2. Show i x j = k , k x i = j , j x k = i ; j i X k = — j , k X j = —i. i. 3. 1. 4. (a) Find a vector perpendicular to the plane containing i + j and 2i - j - k. (b) Find all vectors perpendicular to this plane. 5. Find a vector perpendicular to the plane containing the points (0, 0, 0 ) , (1, 2, 3 ) , and (2, 1, 1). 6. Find a vector perpendicular to the plane containing the points (2, 1, - 1 ) , (1, 0, 1), and (3, 3, 2 ) .

18. Prove that for every two vectors u and v, ||u + v | | 2 + | | u - v | | 2 = 2(||u|p+||v||2). 4. 19. 18 to show that the sum of the squares of the lengths of the four sides of any parallelogram equals the sum of the squares of the lengths of the diagonals. 7 . 1 DEFINITION u X V = (0263 — A362)i + (azbi — aibz)j + {αφι — αφι)}α where the a's are the components of u and the Vs are the com­ ponents of V. Admittedly this arrives at a reasonably important product through the back door. It would have been more in the spirit of our previous definitions if we had defined the cross product by specifying its geometrical properties, independent of compo­ nents or coordinate systems.

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A First Course in Linear Algebra by Daniel Zelinsky and Samuel S. Saslaw (Auth.)


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