Need help on Question 7 and 8, step by step please. Have no clue on these things.

 

 

Homework 2
Math 2210Q, Fall 2016
Dr. Erik Wallace Name:
Due: Monday, September 19
1. Given the matrices


2 0 −1
A=
3 −4 5 −3
B= 4
5 0
1
−2 
C= 1
0 
2
1 D= 
0
3 
−5
0 calculate each of the following if it exists:
AB + C, BA + C, B(C + D),
CD, DC, A(B + C) BC + D, ABC −2D CA, 2. Calculate the product

0
0 1
1 
1
0 1
0  Use this example to show that given AB = AC we cannot conclude B = C (this is known as the law of
cancellation, and it is true for real numbers but not for matrices).
3. Calculate the product

1
0
and find the value of b such that the result is 2
1 
1
0 
1
0 0
1 b
1   4. Find a dependence relation between the columns of

2 0
A=
3 −4 
−1
5 Is any column a multiple of another column?
5. Find a dependence relation between the rows of

21 −28
A=
3
−4 35
5  6. Find a linearly independent subset of the vectors 1
1
1
1
0
−2 v1 = 0 , v2 = 1 , v3 = 3 ,
0
0
0 2
1 v4 = 1 .
1 Can each of the four vectors be written as a linear combination of the other three?
1 7. Suppose a linear system has the solution 2
−1
1
1 + y 1 + z 1
1
0
1
a. What is the dimension of the span?
b. What is a parameterization of the x, y, z coordinates of a vector in the span? 2
c. Is the vector 1 in the span?
1
d. Is the solution set
(a) the same as the span,
(b) parallel to the span,
(c) perpendicular to the span,
(d) none of these.
8. True or False?
a. If 3 vectors lie in the same plane, then there is a dependence relation between them.
b. If a set of vectors is linearly dependent, then a vector in the set is a scalar multiple of one of the others.
c. If a set of vectors is linearly dependent, then each vector in the set can be written as a linear combination
of the others.
d. If a set of vectors is linearly dependent, then there exists a vector in the set that can be written as a
linear combination of the others.
e. The columns of any 4 × 5 matrix are linearly dependent.
f. A set of fewer than n vectors in Rn is linearly independent. 2

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