514-931-8792 (ext. 412)
Consider the vectors, , and , shown below.
Find the graphical result of the following vector operations. Appr
The structure of a methane (CH4) molecule is such that the carbon atom is positioned at the
center of the four hydrogen atoms. The four hydrogen atoms are at positions
A(0, 0, 0), B(1, 0, 1), C(1, 1, 0), D(0, 1, 1).
Linear systems and Gaussian elimination
Using Gaussian elimination, reduce the following system to its echelon form. Identify all free and
leading variables and write its general solution.
3 x 3 y z 2t 10
x 4 y 5 z 3t 5
5 x 5 y 9 z 8t
Using index notation, demonstrate the left-distributivity property:
C(A+ B) = CA + CB.
Demonstration done in notes
A quadrilateral has its vertices at the coordinate points A , B , C , D .
Vector spaces and subspaces
Define the following concepts :
S is a subspace
The subspace S = spancfw_ u , v , w
Linear independence of vectors u , v , w .
The vectors u , v , w are a basis for a subspace S.
Determinants and its applications
A 3 1 13
Consider the matrix
Compute the determinant of A. What values of k will lead to 1 solution, no solution or infinitely
det A 3
Lines and planes
Consider the plane whose equation is 2 x 3 y 6 z 12 . Find a unit vector that is normal to the
2 3 6
7 7 7
The vertices of a triangle are A(1, 2, 1), B(2, 3, 1) and C(4, 2, 5).
u 3 b 2 a
, where and are perpendicular vectors.
c) Find the length of the vector
Let us assume that
are unit vectors that are separated by a 30 degree angle. Reduce each of the
Consider the triangle show below. Use a projection to find the coordinates of point N if it is the point on
the segment AB that is nearest to point C.
Determine whether or not the following statements are true.
Consider a triangle whose vertices are
. Assume that
of the distance between
is the point that is 2/3
and the midpoint of
BC . In the same manner, assume
is the point that is 2/3 of the distanc
Lab 4: Polar Form of Complex Numbers
1. Convert the following numbers to rectangular form.
a) 3 cos 4 + i sin 4
b) 5 cis
c) cos 7 + i sin 7
d) 10 cis 248 (give two decimal places)
2. Convert the following numbers to polar form. Choose t
Lab 5: Lines & Planes in R3
1. True or false? (Note: distinct in this context means not coincident)
a) Two distinct planes perpendicular to a given line are parallel to each other.
b) A plane and a line not in the plane either intersect or are par
a) A triangle is formed by joining two adjacent corners at the
bottom of a cube to the midpoint of the edge at the top of
the opposite face, as in the picture on the right. Use the dot
product to calculate the internal angles of
August 27, 2015
Nature vs Nurture
Answer might have a big impact on social policy
Where is if worth intervening to help kids and families?
Burt sometimes it is hard to know!
Children active in own development
Inborn qualities elicit
Exercises on (R)REF, rank
1. Decide if each of the following matrices is in RREF, REF (bot not RREF), or neither REF nor
1 4 3 0
a) 0 0
0 1 1 7 0 0
9 0 0
d) 0 0 1
0 0 0
0 0 1
2. Reduce the following matrices to RREF.
Exercises on Number of Solutions
1. Give the general solution to the linear system whose augmented matrix has been reduced to the
following. Give the solution in (i) parametric equation form, and (ii) vector form.
1 2 0 10 0 55
0 0 1 5 0 40
Exercises on Elimination
1. Solve the following systems of linear equations by Gaussian elimination with back-substitution.
2x2 4x3 + 13x4 = 11
8x2 + 16x3 49x4 = 45
2x2 3x3 + 14x4 = 6
6x2 14x3 + 35x4 = 44
2. Solve the
201-NYC-05 (Linear Algebra I); A15
Addendum to Course Outline: Evaluation Method
Your evaluation in this course will comprise a series of assignments, two tests and a final
From the course outline, the break-d
Linear Algebra I
Sec. V Math TS or Sec. V
Math SN or equivalent
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