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  EE 501 Linear Algebra and OptimizationAssignment # 3 1. Let  V    = C . Show that   z,w   := Re( z  ¯ w ) defines an inner product on  C .2. Let  V    = R 2 and define   x,y   :=  y 1  (2 x 1  +  x 2 )+ y 2  ( x 1  +  x 2 ). Does this define an inner product on R 2 ?3. Let  f   : [0 , 1]  → R  be continuous with  f  ( t )  ≥  0 for  t  ∈  [0 , 1]. Show that   10  f  ( t ) dt  = 0 if and only if  f  ( t ) = 0 for all  t  ∈  [0 , 1].4. Let  C [0 , 1] be the set of all continuous functions on [0 , 1]. If   f,g  ∈ C [0 , 1], define   f,g   =   10  f  ( t ) g ( t ) dt . Show that   f,g   defines an inner product on  C [0 , 1].5. Let  V    = R n × n . Define   A,B   := Tr( AB T  ). Show that this defines an inner product on  R n × n .6. Fix  a  ∈  V    . Show that the maps from  V    to  R  given by  x  →  x,a   and  y  →  a,y   are linear.7. Let ( v 1 ,...,v n ) be a basis (not necessarily orthonormal) of   R n . Let  α i  ∈  R  be given for 1  ≤  i  ≤  n .Show that there exists a unique vector  x  ∈ R n such that   x,v i   =  α i  for all  i .8. Let  a i ,  1  ≤  i  ≤  n  be positive and real. Let  α  ∈ R . Show that   n  i =1 √  a i  2 ≤   n  i =1 a αi   n  i =1 a 1 − αi  , with equality if and only if either  α  =  12  or  α   =  12  but all the  a i ’s are equal.9. A metric on a set  X   is a function  d  :  X   × X   → R  with the following properties:(a)  d ( x,y )  ≥  0 for  x,y  ∈  X   and  d ( x,y ) = 0 if and only if   x  =  y .(b)  d ( x,y ) =  d ( y,x ) for all  x,y  ∈  X  .(c)  d ( x,z )  ≤  d ( x,y ) +  d ( y,z ) for all  x,y,z  ∈  X  .Let  V    be an inner product space. If we define d ( x,y ) :=   x − y   for  x,y  ∈  V, show that  d  is a metric on  V    .10. Let  x,y,z  ∈  V    . Let  x  ⊥  y  and  x  ⊥  z . Show that  x  ⊥  ( αy  +  βz ) for all  α,β   ∈ R .11. If a vector  z  ∈  V    is orthogonal to all the vectors in  V    , then show that  z  = 0.12. Let  S   be a nonempty subset (not necessarily a subspace) of   V    . Let S  ⊥ :=  { v  ∈  V    |  v,s   = 0 ,  for all  s  ∈  S  } Show that  S  ⊥ is a subspace of   V    .13. Let  v  = ( α,β  )  ∈ R 2 be nonzero. Describe  v ⊥ as span( w ) for a suitable  w .14. Let  v  and  w  be two nonzero vectors in  R 3 . Assume that the set of vectors orthogonal to both of them is a plane through the srcin. Show that each is a scalar multiple of the other.15. Let  v  = ( α,β,γ  ) be a nonzero vector in  R 3 . Find a basis of   W   :=  v ⊥ .16. Show that   x  +  y   =   x  +  y   if and only if one is a nonnegative scalar multiple of the other.17. Prove that   x   =   y   if and only if ( x − y )  ⊥  ( x  +  y ).18. Show that a parallelogram is a rectangle if and only if the diagonals are of equal length.  19. Show that, if a triangle is isosceles, then the medians to the two sides of equal length are of equallength.20. Show that the medians of triangle are concurrent.21. Show that the angle inscribed by semicircle is a right angle.22. Is the standard basis for  R n × n an orthonormal basis?23. Apply Gram-Schmidt process to obtain orthonormal sets:(a)  [ − 1 , 0 , 1] T  , [1 , − 1 , 0] T  , []0 , 0 , 1] T    in  R 3 .(b)  1 ,p 1 ( t ) =  t,p 2 ( t ) =  t 2   of   P  2 ( R ) with the inner product   p,q    :=    10  p ( t ) q  ( t ) dt. (c)  [1 , 1 , 1 , 1] T  , [0 , 2 , 0 , 2] T  , [ − 1 , 1 , 3 , − 1] T    in  R 4 .(d)  [1 , − 1 , 1 , − 1] T  , [5 , 1 , 1 , 1] T  , [2 , 3 , 4 , − 1] T    in  R 4 24. If   V    is an inner product space, a linear map  T   :  V    →  V    is said to be  symmetric   (with respect to thegiven inner product) if    T  ( x ) ,y   =   x,T  ( y )   for all  x,y  ∈  V    . Show that the matrix representing asymmetric map with respect to an orthonormal basis is symmetric.25. Let  V    be an inner product space. A linear transformation  T   :  V    →  V    is said to be  orthogonal   if   T  ( x ) ,T  ( y )   =   x,y   for all  x,y  ∈  V    . Prove that the following are equivalent in the case of a lineartransformation  T   :  V    →  V   (a)  T   is orthogonal.(b)   Tx   =   x   for all  x  ∈  V    .(c)  T   takes an orthonormal basis to an orthonormal basis. That is, if   { e i } ni =1  is an orthonormalbasis, then  { T  ( e i ) } ni =1  is an orthonormal basis.26. If   f   :  V    →  V    is any map such that(a)  f  (0) = 0(b)   f  ( x ) − f  ( y )   =   x − y  ,then show that  f   is an orthogonal linear transformation.27. Let  g  :  V    →  V    be such that   g ( x ) − g ( y )   =   x − y   for all  x,y  ∈  V    . Show that there exists aunique  v  ∈  V    and an orthogonal linear transformation  A  :  V    →  V    such that  g ( x ) =  Ax  +  v .28. Let  V    be an arbitrary vector space. Let  T  v  denote the translation by  v . Prove the following:(a)  T  v  is a bijection for all  v  ∈  V    such that  T  − 1 v  =  T  − v  =  − T  v .(b)  T  v + w  =  T  v  +  T  w  for all  v,w  ∈  V   29. Let  A  be an orthogonal linear map of   V    ,  T  v  a translation. What is the inverse of   AT  v ? What are AT  v ,  T  v A ,  AT  v A − 1 and  T  − 1 v  AT  v ?30. A matrix  A  is orthogonal if and only if   A T  A  =  AA T  =  I  . Show that1 = det  I   = det  AA T    = det  A  det  A T  = (det A ) 2 . 31. With a fixed orthonormal basis, prove that there exists a one-one correspondence between orthog-onal transformations and orthogonal matrices.32. Show that the choice of an orthonormal basis ( v 1 ,...,v n ) for  V    gives rise to an orthogonal linearmap  T   from  V    to  R n :  T  ( v i ) :=  e i  for 1  ≤  i  ≤  n  and extended linearly.2  33. Let A  =   0 11 0  Find the eigenvalues and eigenvectors of   A .34. Find by inspection the eigenvectors and eigenvalues of   1 1 01 1 00 0 1  35. Find the eigenvectors and eigenvalues of   0 0 20 2 02 0 3  36. Let  T   :  V    →  V    be a symmetric linear map. Show that the eigenvectors  v i  with eigenvalues λ i , i  = 1 , 2 with  λ 1   =  λ 2  are orthogonal to each other.37. Show that, if the characteristic equation of   T   has  n  distinct real roots, then  T   is diagonalizable.38. Let  T   :  V    →  V    be a symmetric linear map. Assume that  W   is a vector subspace of   V    invariantunder  T  . Prove that  W  ⊥ is also invariant under  T  .39.  Read a proof of the Spectral Theorem for Symmetric Linear Maps :Let  T   :  V    →  V    be a symmetric linear map on a (finite dimensional real) inner product space. Thenthere exists an orthonormal basis of   V    consisting of eigenvectors of   T  .3
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