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124 Linear Algebra—Practice exam 1 University of Vermont, Spring Semester
Name:
ä Total points: 24 (3 points per question); Time allowed: 75 minutes. ä Brains only: No calculators or other electronic gadgets allowed. ä For full points, please show all working clearly.
- Consider the following system of linear equations expressed in matrix form as A~x = ~b: (^) [ 1 1 − 2 1
] [
x 1 x 2
]
[
]
The solution is ~x =
[
x 1 x 2
]
[
]
Write down the row equations for this problem and draw the ‘row picture.’ Please label your graph thoroughly.
- For the same system given in the preceding question, write down the equation for and draw the ‘column picture.’ Again, label well.
- Solve A~x = ~b for ~x using forward and back substitution given A = LU where
L =
[
]
and U =
[
]
and ~b =
[
]
- Find the inverse of A =
(^) by performing Gauss-Jordan elimination
on [ A | I ]. Clearly indicate the row operations you perform. Smile.
- Given the linear systems of equations
0 h 3 h 0 2 h 2
x 1 x 2 x 3
for which values of h will there be (a) infinitely many solutions, (b) no solutions, and/or (c) one solution?
- Grab bag:
(a) If A is 5 by 17, what shapes are L and U?
(b With what kind of (unusual) elimination matrix Eweirdo would we encode the (unusual) row operation R 3 ′^ = 13 R3 + 2R 2 − R 1 for a 3x3 matrix? (Here, R 1 = Row 1, etc., and it is implied that rows 1 and 2 are unchanged.)
(c) State the rule for expressing (AB)T^ in terms of AT^ and BT, and prove that ATA is symmetric for any matrix A.
