EUS Tutoring: Math 100, Midterm 2 Study Sheet
1. Let:
�⃑:
a. Compute 2𝐴⃑ + 4𝐷
𝐴⃑ = < 3,0,2 >
�⃑ = < −4,1,6 >
𝐵
𝐶⃑ = < 10,9,0 >
�⃑ = < 7,3,5 >
𝐷
�⃑�:
b. Compute�𝐷
�⃑ − 𝐶⃑�:
c. Compute�2𝐵
�⃑ − 𝐴⃑) ∙ (𝐶⃑):
d. Compute (3𝐷
�⃑ :
e. Compute the angle between 𝐴⃑ and 𝐵
2. Consider the augmented matrix below:
1
0
�
2 6
| �
0 3
Determine whether the matrix has one solution, no solution, or an infinite number of
solutions.
3. Consider the augmented matrix below:
1
�0
0
4 5 2
4 2 | 3�
0 9 1
Determine whether the matrix has one solution, no solution, or an infinite number of
solutions.
4. Consider the following lines of MATLAB code:
A = [1 0 0; 3 2 5; 2 3 4];
A = A + [3 2 1; 0 0 0; 1 2 1];
What will be the output in MATLAB when the command “A(2,1) + A(1,2)” is called?
5. Are the vectors A = <1,4> and B = <0,0> linearly independent? Explain.
6. Are the vectors A = <1,2,3>, B = <1,0,0> and C = <0,5,4> linearly independent? Explain.
7. Let vector A = <1,2,3> and B = <2,4,k>. For what values of k are A and B linearly
dependent?
8. A linear system of with four equations and four unknowns has solutions that lie on the
line: <3,0,1,0> + t<5,0,1,2>. What is the reduced row echelon form of the augmented
matrix of this system?
9. Consider the linear systems, with unknowns x, y and z:
𝑥 + 2𝑦 + 3𝑧 = 9
𝑥 − 𝑦 + 2𝑧 = 9
𝑥 + 4𝑦 + 9𝑧 = 0
a) Write the system as an augmented matrix
b) Use Gaussian Elimination on the matrix to reduce it to upper triangular form
c) Find the solution to the system
10. The line L passes through the points (9,0,1) and (7,2,3).
a) Find a parametric equation for L.
b) The place P has the equation:− 𝑥 + 𝑦 + 𝑧 = 5. Is the line L parallel to P,
perpendicular to P, or neither?
c) The plane Q has the equation:2𝑥 − 2𝑧 = 1. Is the line L parallel to Q, perpendicular to
Q, or neither?
d) Find an equation for the plane that is perpendicular to L and passes through the point
(6,2,4).
1. Answer the following true/false questions. Include a short explanation.
a) Elementary row operations on an augmented matrix never change the solution set of
the associated linear system.
b) An inconsistent system has more than one solution.
c) If two vectors have a dot product of 0, then the angle between them is 90 degrees.
d) If one row in an echelon form of an augmented matrix is [0 0 0 0 6], then the
associated linear system is inconsistent.
e) Whenever a system has free variables,the solution set contains many solutions.
f) If two vectors have a cross product of 0, then they are parallel.
g) If the equation Ax = b is inconsistent, then b is not in the set spanned by the columns
of A.