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11hw4

# 11hw4 - Homework Assignment#4 13.12 Write the following...

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Homework Assignment #4 13.12 Write the following quadratic forms in matrix form: a ) x 2 1 - 2 x 1 x 2 + x 2 2 . b ) 5 x 2 1 - 10 x 1 x 2 - x 2 2 . c ) x 2 1 + 2 x 2 2 + 3 x 2 3 + 4 x 1 x 2 - 6 x 1 x 3 + 8 x 2 x 3 . Answer: If we require the matrices to be symmetric, the solutions are: ± 1 - 1 - 1 1 ² , ± 5 - 5 - 5 - 1 ² , 1 2 - 3 2 2 4 - 3 4 3 . 13.21 Let f : R k R 1 be continuous at the point a = ( a 1 ,... ,a k ). Consider the function g : R 1 R 1 deﬁned by g ( t ) = f ( t,a 2 ,... ,a k ). Show that g is continuous at a 1 . This result implies that if f is continuous, its restriction to any line parallel to a coordinate axis is also continuous. However, the converse is not true. Consider the function f ( x,y ) = xy 2 / ( x 2 + y 4 ). Show that f 1 ( t ) = f ( t,a ) and f 2 ( t ) = f ( a,t ) are continuous functions of t for each ﬁxed a . Show that f itself is not continuous at (0 , 0). [Hint: Take a sequence on the diagonal.] Answer: Let t n t . Then ( t n ,a 2 ,... ,a k ) ( t,a 2 ,... ,a k ). Since f is continuous, g ( t n ) = f ( t n ,a 2 ,... ,a k ) f ( t,a 2 ,... ,a k ) = g

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11hw4 - Homework Assignment#4 13.12 Write the following...

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