Question 1 worth 6 points Let H be the set of all polynomials of the form p(t) = a + bt3 where a and b are in .  Determine whether H is a vector space. If it is not a vector space, mention at least one reason why.

Question 5 worth 8 points Given the set of vectors , determine whether the set of vectors is a basis for . Explain your reasoning.

Question 2 worth 8 points If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space. is the set of all vectors of the form, where and are arbitrary real numbers.  

34. On examination, Mrs. Smith has a grade III/IV murmur. This means that the intensity is:

Question 5 worth 10 points Recall that  matrix  is similar to  matrix  if there is an invertible matrix  such that 

Question 9 worth 10 points Consider the subspace H of R3 that has the basis  a.   Find the xyz-equation that defines subspace . b.   Using the given basis , find an orthonormal basis for  (and verify!) c.   If 

Question 12 worth 9 points An ice-cream shop sells only ice-cream sodas and milk shakes. It puts: 1 ounce of syrup and 4 ounces of ice-cream in an ice-cream soda 1 ounce of syrup and 3 ounces of ice-cream in a milk shake   If the store used 160 ounces of syrup and 512 ounces of ice-cream, how many ice-cream sodas and milk shakes did it sell? a)  Set up the system of equations, letting: x = # ice-cream sodas sold y = # milk shakes sold   b)  Solve the system (show your work and justify your answer).  

Question 6 worth  8 points Let :

Question 5 worth 10 points Recall that   matrix  is similar to  matrix  if there is an invertible matrix  such that 

Question 6 worth 10 points Orthogonally diagonalize the given symmetric matrix  by finding an orthogonal matrix  and diagonal matrix  such that