linear-guest

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 equations.
The change in voltage (measured in Volts) around any loop due to
batteries | ∣ ∣ and resistors /\/\/\/\ (given by the product of the current measured
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
60 Volts
13 Amps
80 Volts
J Amps
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
Also assume
m ≠ n .
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?
(d) What size matrix is M T M?
(e) What size matrix is MM T ?
(f) Is M T M invertible?
(g) is M T M symmetric?
342