Question 4 worth 8 points Consider the polynomials: p1(t) = 2 + t p2(t) = -2t p3(t) = 1 (i) Find a linear dependence relation among p1, p2, p3. (ii) Find a basis for Span {p1, p2, p3}. (iii) Use your answer from part (ii) to express v(t) = 6 + 4t  as a linear combination of vectors from the basis.    

Question 4 worth 8 points Consider the polynomials: p1(t) = -t p2(t) = 2 + 2t p3(t) = -4 (i) Find a linear dependence relation among p1, p2, p3. (ii) Find a basis for Span {p1, p2, p3}. (iii) Use your answer from part (ii) to express v(t) = 6 + 4t  as a linear combination of vectors from the basis.    

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Question 4 worth 10 points Consider the probability matrix , where 

Question 3 worth 10 points Determine whether the given matrix A is diagonalizable. If it is, find an invertible matrix  such that 

Question 1 worth 10 points Given A=            a.   Find matrices 

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 7 worth 10 points Consider the linear transformation :

Question 4 worth 10 points Suppose matrix  has eigenvalues 

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