Page 342 - 35Linear Algebra

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 ?
T
(f) Is M M invertible?
T
(g) is M M symmetric?

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