MECE 6384
Assignment 1
Fall 2022
Solutions will be posted on Friday, September 2.
Due Date for submission of graded assignment: Tuesday, September 6
Each question carries 10 points.
1. True of False (Justify your answer):
a) The set of all vectors in R3 satisfying −v1 + 2v2 + 3v3 = 0, −4v1 + v2 + v3 is a linear
vector space over R.
b) The set of all polynomials in x of degree 3 or less, is a linear vector space over R.
c) The set of all non-singular N × N matrices form a linear vector space of dimension
N 2 over R.
d) the set of all functions y(x) = (ax + b)e−x with any coefficients a and b forms a
linear vector space over R.
2. What restriction must be made on b so that the vector 2e1 + be2 is orthogonal to (a)
−3e1 + 2e2 + e3 and (b) e3 , where e1 , e2 , and e3 are orthonormal basis vectors in R3 .
3. For the given matrices, compute the expressions or give reason why they are not defined:




 
3 1 −3
0 4 1
2
A =  1 4 2  , B = −4 0 −2 , u =  0 
−3 2 5
−1 2 0
−5
a) AB − BA; b) uT Au; c) det A2
4. Determine the rank of the following matrices:
4 −2 6
−2 1 −3


0 3 5
, 3 5 0 
5 0 10
5. Solve the following simultaneous equations for x1 , x2 , x3 using matrix methods:
x1 + 2x2 + 3x3 = 1
3x1 + 4x2 + 5x3 = 2
x1 + 3x2 + 4x3 = 3
6. Show that the following equations have solutions only if η = 1 or 2, and find the
solutions in these cases:
x+y+z =1
x + 2y + 4z = η
x + 4y + 10z = η 2
7. Find the inverse transformation:
y1 = 0.5x1 − 0.5x2
y2 = 1.5x1 − 2.5x2
8. A and B are real non-zero 3 × 3 matrices that satisfy the equation
(AB)T + B−1 A = 0
Prove that if B is orthogonal, then A is skew-symmetric.