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moienMath 21b: Linear Algebra Spring 2016
Homework 1: Linear Equations
This homework is due on Monday, February 1, respectively Tuesday February 2, 2016. Homework
is due at the beginning of each class in the classroom.
2x + 3y + 3z = 39
1 Find all solutions of the linear system x + 2y + 3z = 32
x + y + 4z = 35
2 Jake and Maggie are siblings. Maggie has twice
as many brothers as sisters and Jake has just as
many brothers as sisters. How many kids are
there in this family? (Any resemblance to the
Gyllenhaal siblings is purely coincidental).
3 A 10 km trip from the Swiss waterfall “Rhein-
fall” to the village “Rheinau” takes 30 minutes.
The return trip takes an hour. How fast is the
speed v (in km/h) of the boat traveling relative
to the water, and how fast is the speed s (in
km/h) of the river?
4 On a heating mesh, the temperature at exterior
mesh points is 0, 200 or 400 F as given in the 0
picture. In thermal equilibrium, each interior 200 T1 0
mesh point has the average of the temperatures
200 T2 T3 0
at the 4 adjacent points. For example T2 =
(T3 + T1 + 200 + 0)/4. Find the temperatures 0 400
T1, T2, T3.
5 A polyhedron has v vertices, e edges and f tri-
angular faces. Euler proved his famous formula
v − e + f = 2. There is an other relation,
3f = 2e called a Dehn-Sommerville relation
which always holds. The number f is called
the area. You get a polyhedron with area 288.
Write down a system of equations in matrix form
Ax = b. Then determine the number of vertices
and edges.
Main definitions
A linear equation for finitely many variables
x1, x2, . . . , xn is an equation of the form
a1x1 + a2x2 + · · · + anxn = b .
If m equations of this type are given, we have a system of
linear equations. It can be written in matrix form A~x =
~b, where ~x is a column vector containing the n variables and
the m × n matrix A lists all the m · n coefficients. Finally,
~b is the column vector listing the values to the right. For
example, the system x + 2y + z = 8, 3x − y − 7z = 4 of
equations can be written as
x
1 2 1 8
y = .
3 −1 −7 4
z
Unlike in the homework examples given here, the number of
variables and equations do not always have to be the same.