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**Unformatted text preview: **Math 375 Review problems II 1. Let V be the vector space of all vectors x R 4 with the property x 1 + x 2 + x 3 + x 4 = 0 . Then u 1 = (1 , , ,- 1) and u 2 = (0 , 1 ,- 1 , 0) are two linearly independent vectors in V . (i) Find an orthonormal basis of the span of u 1 , u 2 . Extend it to an orthonormal basis of V . Then extend this basis of V to an orthonormal basis of R 4 . (ii) Let x = (1 , , 1 , 1) and let d = min a V k a- x k . Determine d and a vector b V so that k b- x k = d . 2. (i) Let f 1 , f 2 and f 3 be three real-valued integrable functions which satisfy Z 1 [ f 1 ( x )] 2 dx = 2 , Z 1 [ f 2 ( x )] 2 dx = 5 , Z 1 [ f 3 ( x )] 2 dx = 5 , Z 1 f 1 ( x ) f 2 ( x ) dx = Z 1 f 1 ( x ) f 3 ( x ) dx = Z 1 f 2 ( x ) f 3 ( x ) dx = 0 Z 1 xf 1 ( x ) = 1 , Z 1 xf 2 ( x ) dx = 0 , Z 1 xf 3 ( x ) dx = 2 Find the minimum of the expression Z 1 | x- 3 X i =1 i f i ( x ) | 2 dx and determine the coefficients 1 , 2 , 3 for which this minimum is attained (ii) Find coefficients a , a 1 , ..., a N which minimize the expression Z x- N X k =0 a k cos kx 2 dx. 3. Consider the linear transformation R 5 R 3 given by T ( x ) = x 1 + x 2 + x 4 + x 5 x 2 + x 3 + x 4 + x 5 2 x 1 + 3 x 2 + x 3 + x 4 + x 5 (i) Find a basis of the nullspace of T . (ii) Find a basis of the range of T . (iii) Give a complete description of the set of solutions of the linear system x 1 + x 2 + x 4 + x 5 = b 1 x 2 + x 3 + x 4 + x 5 = b 2 2 x 1 + 3 x 2 + x 3 + x 4 + x 5 = b 3 , depending on ( b 1 , b...

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