MATH 110, Fall 2014
Tutorial #4
October 1, 2014
Todays main problems
Consider the vectors
1
2
a=
1
0
1
1
b=
0
1
3
4
c=
1
2
1
3
d=
2
1
1. Is the set cfw_a, b, c, d linearly independent?
2. Describe spancfw_a, b, c, d as a point, line, plane, or h
MATH 110, Fall 2014
Tutorial #3
September 24, 2014
Todays main problems
1. Consider the system
x 3y 5z = 0
y + z = 3.
(a) Row reduce the corresponding augmented matrix.
(b) Identify the solution as a point, line, or plane.
(c) Write the solution in vector
MATH 110, Fall 2014
Tutorial #2
September 17, 2014
Todays main problems
The line L is dened by the equation y = 2x + 3.
1. Find a constant k so that the equation
2y + kx = 6
also describes L
2. Write L in vector form and write down a normal vector for L.
MATH 110, Fall 2014
Tutorial #9
November 19, 2014
Todays main problems
1
2 1
1. Consider the matrix F = 1 1 3
4
0 k
(a) Compute det(F ).
(b) For what values of k is F invertible?
(c) For what values of k is zero an eigenvalue of F ?
2. Consider the matrix
MATH 110, Fall 2014
Tutorial #11
November 24, 2014
Todays main problems
V = cfw_v1 , v2 , v3 where
0
2
v1 =
0
4
0
v2 =
3
3
0
v3 =
4
1. (a) Show that V is an orthogonal basis. Is it an orthonormal basis?
(b) Create an orthonormal basis B = cfw_b1 ,
MATH 110, Fall 2014
Tutorial #9
November 5, 2014
Todays main problems
1. The transformation S : R2 R2 is given by S
x
2x
=
is linear.
y
y
(a) Find the standard matrix representation for the transformation S.
(b) Explain why this linear transformation is c
MATHEMATICS 110, Fall 2014,
Tutorial #1
September 11, 2014
Exercise 1
3
-2
For the vectors u = -1 and v = 1 determine:
4
-2
(A) the distance between the vectors u and v;
(B) the unit vector in the direction of the vector u;
(C) the projection of the vecto
MATH 110, Fall 2014
Tutorial #6
October 15, 2014
Todays main problems
1 2 3
1. Find the inverse of A = 2 5 3 if it exists or show that it is not invertible.
1 0 8
2. Using Problem 1, solve the system
x + 2y + 3z = 3
2x + 5y + 3z = 1
x + 8z = 3.
3. For the
MATHEMATICS 110, Fall 2014,
Tutorial #7
Oct 22, 2014
1
Todays main problems
1 1 0 3
2 5 2 6
Exercise 1 Compute the determinant of A =
0 1 0 0.
1 2 3 4
x1
2
3
Exercise 2 Compute the determinant of A = 2
x2
4 , where x R.
3
4
x3
1 x x2
Exercise 3 Compute t
MATH 110, Fall 2014
Tutorial #12
December 3, 2014
Todays main problems
Consider the set of vectors V = cfw_v1 , v2 , v3 , v4 , where
0
1
1
2
1
2
v1 =
v2 =
v3 =
1
0
0
0
0
1
1
0
v4 = .
0
1
1. Apply the Gram-Schmidt process to V to nd an orthonorm
MATHEMATICS 110, Fall 2014,
Tutorial #7
Oct 22, 2014
1
Todays main problems
1 1 0 3
2 5 2 6
Exercise 1 Compute the determinant of A =
0 1 0 0.
1 2 3 4
x1
2
3
Exercise 2 Compute the determinant of A = 2
x2
4 , where x R.
3
4
x3
1 x x2
Exercise 3 Compute t
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MATHEMATICS 110, Fall 2014,
Tutorial #1
September 11, 2014
Exercise 1
3
-2
For the vectors u = -1 and v = 1 determine:
4
-2
(A) the distance between the vectors u and v;
(B) the unit vector in the direction of the vector u;
(C) the projection of the vecto
MATH 110, Fall 2014
Tutorial #6
October 15, 2014
Todays main problems
1 2 3
1. Find the inverse of A = 2 5 3 if it exists or show that it is not invertible.
1 0 8
2. Using Problem 1, solve the system
x + 2y + 3z = 3
2x + 5y + 3z = 1
x + 8z = 3.
3. For the
MATH 110, Fall 2014
Tutorial #12
December 3, 2014
Todays main problems
Consider the set of vectors V = cfw_v1 , v2 , v3 , v4 , where
0
1
1
2
1
2
v1 =
v2 =
v3 =
1
0
0
0
0
1
1
0
v4 = .
0
1
1. Apply the Gram-Schmidt process to V to nd an orthonorm