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Name:
Instructor:
NORTHEASTERN UNIVERSITY
Department of Mathematics
MTH 1230
FINAL EXAM Fall 1997
Instructions: Put your name and the instructor's
name in the blanks above. Put your final answers to each question
in the designated spaces--you may lose credit if you don't.
Calculators are permitted. A single cribsheet of formulas
is allowed. SHOW YOUR WORK. If there is not
enough room to show your work, use the back of the preceding page.
Good luck!
- 8 points
Are the following vectors independent or dependent?
If they are independent, say why. If they are dependent,
exhibit a linear dependence relation among them.
- 20 points
Let
,
.
- Solve the system of linear equations Ax=b, indicating clearly
the row operations, pivots, leading variables, and free variables.
- Find the LU-decomposition of the matrix A.
- Find a basis for each of the row space, column space,
and null-space of A.
- Find the rank of A.
- Find a non-zero vector that is orthogonal to both the
vectors and
.
- 12 points
In each of the following, a vector space V and a
subset S are given. Circle one answer:
- ,
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S is closed under addition: |
YES |
NO |
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S is closed under scalar multiplication: |
YES |
NO |
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S is a vector subspace of V: |
YES |
NO |
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- ,
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S is closed under addition: |
YES |
NO |
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S is closed under scalar multiplication: |
YES |
NO |
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S is a vector subspace of V: |
YES |
NO |
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- ,
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S is closed under addition: |
YES |
NO |
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S is closed under scalar multiplication: |
YES |
NO |
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S is a vector subspace of V: |
YES |
NO |
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- ,
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S is closed under addition: |
YES |
NO |
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S is closed under scalar multiplication: |
YES |
NO |
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S is a vector subspace of V: |
YES |
NO |
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- 12 pts
- Find the matrix A associated with the linear
mapping that rotates the yz-plane by
and reflects the x-axis about the yz-plane.
- Is A orthogonal?
- What is ?
- What is the image of the point (-3,2,1) under the above mapping?
- 12 pts
Let
and .
- Find the least squares solution
of the inconsistent system
.
- Use your answer to part one to find the projection of
onto , the column space of A.
- 16 pts Let
- Find unit vectors in the direction of and ,
respectively.
- Find the lengths of and , and compute
the dot product .
- Find the angle between and . Are
and orthogonal?
- Let .
Use the Gram-Schmidt process to find the QR-factorization of A.
- 20 pts
Let
- Find the characteristic equation for A.
- Find the eigenvalues of A.
- Find a basis for each eigenspace of A.
- Form a matrix S using the two independent eigenvectors
from part c as column vectors, and calculate S-1.
- Calculate . Explain your answer.
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Alexandru I. Suciu
12/8/1997