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Linear Independence and System of Linear Equations in Matrix Form, Exercises of Linear Algebra

A math quiz focused on linear independence of vectors and solving systems of linear equations using matrix form. It includes defining linear independence, finding the reduced row echelon form (rref) of a matrix, and using rref to determine if a system of linear equations has a solution and finding that solution.

Typology: Exercises

2012/2013

Uploaded on 02/27/2013

senapathy_101
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Math 205B Quiz 03 page 1 10/02/2009 Name
1. Let Sbe a set {v1,v2, . . . , vp}of vectors in Rm. Define what it means to say the set Sis linearly independent.
2. Let A=
1 6 3 3 1
1 7 3 3 1
2 13 7 9 4
1 6 5 9 5
and label its column vectors as c1,c2, . . . , c5. It’s a fact that the RREF of
1 6 3 3 1 1 0 0 0
1 7 3 3 1 0 1 0 0
2 13 7 9 4 0 0 1 0
1 6 5 9 5 0 0 0 1
is
10065 0 23 17 10
0 1 0 0 0 0 3 2 1
0 0 1 3 2 0 1 1 1
0 0 0 0 0 1 2 2 1
.
2A. Let b=
b1
b2
b3
b4
. Use the above information to decide what conditions, if any, b1,b2,b3, and b4must satisfy in order
for Ax=bto have a solution.
2B. Without doing any work, find an easy solution of Ax=c3.
2C. Since Ax=c3does have a solution, the entries of c3must satisfy the conditions in (2A). Show that this is so.
3. Let Aand c1,c2, . . . , c5be as in problem (2).
3A. Is the set S={c1,c2, . . . , c5}linearly independent? Explain.
3B. Show that it is possible to write c3as a linear combination of the other four vectors. (Give an explicit LC).
3C. Explain why it is impossible to write c2as a linear combination of the other four vectors.

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Math 205B Quiz 03 page 1 10/02/2009 Name

  1. Let S be a set {v 1 , v 2 ,... , vp} of vectors in R m . Define what it means to say the set S is linearly independent.
  2. Let A =

and label its column vectors as c 1 , c 2 ,... , c 5. It’s a fact that the RREF of

is

2A. Let b =

b 1

b 2

b 3

b 4

. Use the above information to decide what conditions, if any, b 1 , b 2 , b 3 , and b 4 must satisfy in order

for Ax = b to have a solution.

2B. Without doing any work, find an easy solution of Ax = c 3.

2C. Since Ax = c 3 does have a solution, the entries of c 3 must satisfy the conditions in (2A). Show that this is so.

  1. Let A and c 1 , c 2 ,... , c 5 be as in problem (2).

3A. Is the set S = {c 1 , c 2 ,... , c 5 } linearly independent? Explain.

3B. Show that it is possible to write c 3 as a linear combination of the other four vectors. (Give an explicit LC).

3C. Explain why it is impossible to write c 2 as a linear combination of the other four vectors.