Math 215
Linear Algebra
Spring 2011
Lectures
Feb. 21
Matrix operations; inverse of a matrix (2.1)
Feb. 23
Invertible matrices (2.2)
Feb. 25
Characterizations of invertible matrices (2.3)
Feb. 28
Application to Leontief economic models (2.6)
March 2
Leonti
Math 215 Spring 2015
Problem Set 5 Solutions
1. Dene a function f : cfw_1, 2, 3, 4, . . . , 12 N by letting f (n) be the number of positive
divisors of n. So f (4) = 3 since the divisors of 4 are cfw_1, 2, 4.
(a) Write f as a subset of a Cartesian product
Math 215 Spring 2015
Problem Set 4 Solutions
1. (5 points) For this problem we will do a double containment proof to show the two
sets A = cfw_3x + 1 : x R and B = cfw_3x 2 : x R are equal. Like on the last
homework, some blanks have been left. Fill in ea
Math 215 Spring 2015
Problem Set 1 Solutions
1. (6 points) Write each of the lines described below as a set cfw_c v + w : c is in R (i.e.
ll in v and w in the set notation above).
2
(a) The line through the point (1, 2, 8) and in the direction of vector
Math 215 Spring 2015
Problem Set 2 Solutions
1. Negate the following statements. You do not need to prove or disprove the statements, simply write the negation so that no not appears.
(a) For all x R, we have x3 x2 .
The negation will be:
Not For all x R,
Math 215 Spring 2015
Problem Set 13 Solutions
1. Suppose T : R2 R2 is the linear transformation with [T ] =
and u2 =
2
0
2 1
. Let u1 =
5 2
1
3
.
(a) Prove that Span(u1 , u2 ) = R2 .
Recall from Theorem 2.2.10 that if ad bc = 0 then Span(u1 , u2 ) = R2 .
Math 215 Spring 2015
Problem Set 15 Solutions
3 1
diagonalizable? If so, nd
1 1
some basis = (u1 , u2 ) so that [T ] is a diagonal matrix.
1. Is the transformation T : R2 R2 with [T ] =
No. Refer back to homework 14 where we computed that 2 is the only ei
Math 215 Spring 2015
Problem Set 10 Solutions
1. (a) Write the standard matrix for the transformation which represents projection
onto the line y = 2x.
1
2
Here w =
. So we apply the formula for [Pw ] =
3
2
(b) Use (a) to compute where the vector
Pw 3 ).
Math 215 Spring 2015
Problem Set 9
Due: February 23, 2015
Make sure you are familiar with the Academic Honesty policies for this class, as detailed on the
syllabus. All work is due on the given day by the time lecture starts.
1. Prove Prop 2.3.8(3): Let T
Math 215 Spring 2015
Problem Set 12 Solutions
1. Determine Null(Pw ) where w =
1
1
. (Recall Pw is projection onto the vector w.)
The Null space is all vectors v so that Pw (v) = 0. Consider an arbitrary vector v =
By the formula we developed in class, [P
Math 215
Linear Algebra
Spring 2011
Lectures
Jan. 24
Systems of linear equations; matrix reduction (1.1)
Jan. 26
Row reduction; echelon forms (1.2)
Jan. 28
Vectors in Rn; relationship to systems of equations (1.3)
Jan. 31
The equation Ax=b; geometry and a
Math 215 Spring 2015
Problem Set 6 Solutions
1. (a) Find an example of some u R2 so that Span(u) is the solution set of the
equation 4x 7y = 0.
Algebraically, the solution set of the equation 4x 7y = 0 is given as the line cfw_c
But by the denition of Sp