Question

7. OR. If you prefer, calculate Proj_CS(A)b using the approach of Example 8 on page 266 instead. You should get an infinite number of solutions for x to the normal equation. Any solution for x will work, and then calculate Ax to get Proj_CS(A)
   • For either approach, briefly explain why Proj_CS(A)b is the closest vector in your plane to the vector (See Theorem 5.15 if needed.)
8. Choose 5 arbitrary unique vectors in R² that are not on the same line or scalar multiples of each other. What are your 5 vectors in R²?
   • Use normal equations (see page 265, if needed) to determine the equation of the least square regression line for this set of 5 points.
9. Choose any 2 of the 5 vectors in the above problem that will form a nonstandard basis for R², which we will call B’. (12 points for entire problem)
   • What are your two vectors in B’?
   • If B is the standard basis for R² (I.e. B ={(1.0). (0.1)}, determine the transition matrix (called P^-1 in our text) from B to B’.
   • Use P^-1 to calculate the coordinates for (1,5) with respect to the basis B’.
10. Investigate the following linear differential equation: y’’ + 4y = 0; Solutions {sin(2x), cos(2x)}

• Verify that each solution satisfies the differential equation.
• Use the Wronskian to verify that the solution set is linearly independent.
• Write the general solution of the differential equation.

Answer 1