By Harry Hochstadt
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Extra info for Differential equations : a modern approach
I'M ) ] - ( 5 * (* ). y (*)) + + am ± (JT (* ), ¿ y (» )). 7 27 NORMS OF MATRICES In dealing with ordinary numbers, we often refer to their absolute value or magnitude. This absolute value has the following three properties: (1) i x —y | ^ 0, and vanishes if and only \i x = y. (2) | * — y | = Iy — x |. (3) | * — z \ < | * — y | + \ y — z \ . Such a function is often known as a distance function, since \ x —y \ denotes the distance between the points x and y on a Euclidian line. One can define such distance functions in vector spaces because the function z* \ 1/2 \ X - Y \ = ( X - Y , X - Y)1'2 = ( X ( * * - ^ ) 2) Xi=1 for real vectors X and Y can easily be shown to have properties (1), (2), and (3).
7. Complete the following sets of vectors to obtain bases in a four-dimensional space: 34 LINEAR ALGEBRA 8 . Compute (X , Y) for the following: (a) X = (b) X = (c) X = 9. Represent the vector X = | 0 j in terms of the orthonormal basis vectors This is known as the Gram-Schmidt process. 12* Consider the following matrices: /1 0 = (° 3 \1 0 c = 0 . 0/ B = /2 0 \o G 3). Find D - C, D + C, ! A y C - DTy A B , B A t (CD)Ty (DC)T. 13. Construct orthonormal bases from the bases constructed in problem 7. 14. If find B-'A B . 15. Find a matrix such that A 2 = /. 16. Find matrices corresponding to the following linear transformations. Interpret these transformations geometrically. (a) u = 3xy v = 3y. (b) u = —y, v = x. x —y _ X + y (c) u W T > V 2 * 17.
This is known as the Gram-Schmidt process. 12* Consider the following matrices: /1 0 = (° 3 \1 0 c = 0 . 0/ B = /2 0 \o G 3). Find D - C, D + C, ! A y C - DTy A B , B A t (CD)Ty (DC)T. 13. Construct orthonormal bases from the bases constructed in problem 7. 14. If find B-'A B . 15. Find a matrix such that A 2 = /. 16. Find matrices corresponding to the following linear transformations. Interpret these transformations geometrically. (a) u = 3xy v = 3y. (b) u = —y, v = x. x —y _ X + y (c) u W T > V 2 * 17.