linear-guest

350 Sample Final Exam
(a) The columns of M span in the basis given for .
(b) The rows of M span in the basis given for .
Suppose
⎛
⎞
1 2 1 3
M = ⎜2 1 −1 2
⎟
⎝1 0 0 −1⎠
4 1 −1 0
is the matrix of L in the bases {v 1 , v 2 , v 3 , v 4 } for V and {w 1 , w 2 , w 3 , w 4 }
for W . Find bases for kerL and imL. Use the dimension formula to check
your result.
17. Captain Conundrum collects the following data set
y x
5 −2
2 −1
0 1
3 2
which he believes to be well-approximated by a parabola
y = ax 2 + bx + c .
(a) Write down a system of four linear equations for the unknown coefficients
a, b and c.
(b) Write the augmented matrix for this system of equations.
(c) Find the reduced row echelon form for this augmented matrix.
(d) Are there any solutions to this system?
(e) Find the least squares solution to the system.
(f) What value does Captain Conundrum predict for y when x = 2?
18. Suppose you have collected the following data for an experiment
x y
x 1 y 1
x 2 y 2
x 3 y 3
and believe that the result is well modeled by a straight line
350
y = mx + b .