Linear Algebra
Final Examination
Duration: 110 minutes
Date: June 2, 2020,
Section C
YOUR NAME:
1 (12 points). Find the projection and component of vector A = (0, 4, −3, 2) along
vector B = (7, 1, 2, −1).
2 (13 points). Find the equation of the plane perpendicular to N = (2, 1, −1) and
through the point P = (1, 4, −2).
3 (25 points). Solve (that is find all solutions of) the following system of linear
equations by Gauss-Jordan elimination.
2x − y
x + y
y
x − y
− z − w
+ 7z + w
− z + w
− 9z − w
= −4
=
7
=
6
= −5
Which variables are basic and which are free? Explain.
4 (15 points). What is the dimension of the vector space of 2 × 3 matrices of the
following form
a b 0
.
0 c d
Explain.
5 (17 points). Let L : R3 → R2 be a linear mapping and let
L(−2, 0, 1) = (1, 1), L(1, 2, 0) = (0, 1) and L(3, 1, 5) = (1, 1).
Find L(−1, 1, 1).
6 (18 points). Can the Cramer’s Rule be used to solve the following system of linear
equations? Explain.
2x + y − 4z = −3
x + y + 2z =
4
5x + y + 2z =
0
If ‘yes’ solve the system by this method.