Question 1:

For the system: x + y = 4
                          2x - 2y = 4

Sketch the graph for the row picture.

Sketch the graph for the column picture.

Question 2:

Find the LU factorization of A

Solve L_c=b to find c, then solve Ux = c to find x.

Question 3:

Identify a vector that lies in each of the four fundamental spaces of the system below, prove your choice is an element of the space, and use these to illustrate the relationships between pairs of the fundamental spaces.

x + 2y - 2z = b₁
2x + 5y - 4z = b₂
4x + 9y - 8z = b₃

Question 4:

If addition in R² adds an extra 1 to each component so that and scalar multiplication remains unchanged which of the 8 requirements of a vector space fail to hold?

Question 5:

Show that the set of upper triangular 3 x 3 matrices forms a subspace.

Question 6:

Find solvability condition, the particular solution(s) the special solution(s), and the complete solution for the following system.

1u + 2v + 3w + 5z = 0
2u + 4v + 8w + 12z = 6
3u + 6v + 7w + 13z = -6

Question 7:

Write a matrix that calculates the slope of a linear equation and show that the resulting transformation is linear.

Question 8:

Use Gram-Schmidt to find an orthogonal basis (not normalized) for the vectors

Question 9: Find the equation y = C + Dx of the least-squares line that best fits the data points (2,1); (5,2); (7,3).

Question 10:

Use Cramer's rule to solve the system:

u + 3v = 0
2u + 4v = 6