MATH 200 - LINEAR ALGEBRA HOMEWORK 5
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MATH 200 - LINEAR ALGEBRA
HOMEWORK 5, DUE Monday, November 20, 2023, at 11:59pm
Upload your written (or typed) solutions for this assignment to the appropriate place in the Spring by the due date. You may work on this assignment with your classmates, but you must write up your own solutions.
Upload your Mathematica input and output to the Spring as well, either as a pdf file or an nb file.
Part A. (10 points)
Problem 1. Let
A = l1(a) a(0) 
4 」.
[0 1 a − 5l
Determine all values of a for which the equation A⃗x =0(⃗) has more than one solution.
Problem 2. Let
B = l 1(2) 
2
[ −3 b 3」. 1 l
(a) Determine all values of b such that Col B = R3 .
(b) If Col B 
R3 , find abasis for Col B .
Problem 3. Find a basis for the subspace of R4 consisting of the set of solutions to the system
of equations:
! 〈 (
−x1
3x1
2x1
− x2 − 3x3 − 14x4 = 0
+ 3x2 + 6x3 + 12x4 = 0
+ 2x2 + 4x3 + 15x4 = 0
Part B. (10 points)
The following two Mathematica Commands will be useful for this assignment:
• To find the determinant of a matrix in Mathematica, use the Det command. For example,
to find the determinant of 
3 2(1)], use the command
Det[{{2,1}, {-3,2}}]
• To find the area of a polygon, use the Area command. For example, to find the area of the unit square with vertices (0, 0), (1, 0), (1, 1), and (0, 1), use the command
Area[Polygon[{{0,0},{0,1},{1,1},{1,0}}]]
Problem 1. Let A = I1(3) 4(2)], and let T : R2 → R2 be the linear transformation whose standard
matrix is A.
(a) Use Mathematica to find the determinant of A.
(b) Use Mathematica to plot the unit square S and its image T(S) under the transformation S on the same axes. (Hint: Follow the example in Homework 3.)
(c) Use Mathematica to find the area of the parallelogram T(S).
(d) Compare your answers to (a) and (c). Make a conjecture about the area of T(S) for an arbitrary linear transformation T : R2 → R2 .
Problem 2. Let A = 
2 3(3)] and let T : R2 → R2 be the linear transformation whose standard
matrix is A.
(a) Use Mathematica to compute the determinant of A.
(b) Let P be the parallelogram whose vertices are (0, 0), (4, 1), (5, 2), and (1, 1). Use the methods of Homework 3 to plot P and T(P) on the same axes.
(c) Use Mathematica to compute the area of P and T(P).
(d) Compare your answers to (a) and (c). Make a conjecture about the area of T(P) for an arbitrary linear transformation T : R2 → R2 and parallelogram P.
2023-11-21