Page 342 - 35Linear Algebra
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342 Sample Final Exam
(q) The orthogonal complement to a subspace of a vector space.
2. Kirchoff’s laws: Electrical circuits are easy to analyze using systems of equa-
tions. The change in voltage (measured in Volts) around any loop due to
batteries | and resistors /\/\/\/\ (given by the product of the current mea-
sured in Amps and resistance measured in Ohms) equals zero. Also, the sum
of currents entering any junction vanishes. Consider the circuit
1 Ohm 2 Ohms
I Amps 13 Amps J Amps
60 Volts 80 Volts V Volts
3 Ohms 3 Ohms
Find all possible equations for the unknowns I, J and V and then solve for
I, J and V . Give your answers with correct units.
3. Suppose M is the matrix of a linear transformation
L : U → V
and the vector spaces U and V have dimensions
dim U = n , dim V = m ,
and
m 6= n .
Also assume
kerL = {0 U } .
(a) How many rows does M have?
(b) How many columns does M have?
(c) Are the columns of M linearly independent?
T
(d) What size matrix is M M?
T
(e) What size matrix is MM ?
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(f) Is M M invertible?
T
(g) is M M symmetric?
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