I don't know how to do all of the questions on the hw2. Can you show me the answer and calculation? Thanks

 

 

1. HW 2
1. Suppose {v1 , v2 , v3 , v4 } is a set of linearly dependent vectors.
a. Suppose we apply the elementary operation of adding k times v1 to v3 . Show
the resulting set: {v1 , v2 , kv1 + v3 , v4 } is still linearly dependent.
b. Suppose k 6= 0 and we perform the elementary operation of multiplying v2 by
k. Show the resulting set: {v1 , kv2 , v3 , v4 } is still linearly dependent. 1 1 1 1
1 1 −1 3 2a. Let M = 2 0 1 1.
1 1 2 0
Use column operations to get M as close
to the identity Then as possible. matrix 1
2
2
2
3
4
3
4 determine for which of the vectors b1 = 3, b2 = 4, b3 = 3, b4 = 4,
0
0
1
1
the equation M x = bi has a solution. (You don’t need to solve for x.)


2 −3
2b. Let M =
.
5 4
Use column operations to get M as close
matrix
Then
 to the identity
 
  as possible.
 
1
2
2
1
determine for which of the vectors b1 =
, b2 =
, b3 =
, b4 =
,
2
3
1
3
the equation M x = bi has a solution. (You don’t need to solve for x.)
3. R3 is what we call the vector space of vectors of length 3 with real entries.
Which of the following sets is a basis of R3 ? 1
0
0
a. 0 , 2 , 0
0
0
3 1
0
1
b. −1 , 1 , 0 0
−1
−1 1
1
1
c. 1 , 1 , 0
1
0
0 0
1
d. 0 , 2
1
0
1 2 13
7
3
2
e. 1 , 9 , 1 , 13
6
7
3
2 1
0
0
f. 0 , 2 , 0
0
0
3 0
0
0
1
0 1 0 0 g. 0 , 0 , 1 , 0
1
0
0
0
4. Suppose M is upper diagonal.
a. What must you know about the entries of the diagonal itself in order to be
sure the equation M x = b has a solution for all b.
b. If the entries on the diagonal DO NOT satisfy the condition you stated in
part a), is it still possible for the equation M x = b to have a solution for all b?

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Status NEW Posted 20 Apr 2017 07:04 AM My Price 9.00

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