math 3c practice final problems solutions prepared by vince zaccone for campus learning assistance...
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Math 3CPractice Final Problems
Solutions
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
1a) (find a general solution)
7y2y
First we solve the homogeneous equation:
t2h Cey
...egrateint...separate
0y2y
Now find a particular solution – it will be a constant, say yp=A.
Plug this into the equation and solve for A:
27t2
general
phgeneral
27
p
27
Cey
yyy
y
A
7A20
7)A(2)A(
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
1b) (find a general solution)
tyt
4y
ty4yt 2
First we solve the homogeneous equation:
4Ctln
h
tdt
ydy
tdt
ydy
t4
dtdy
t4
t
Cey
Ctln4yln
4
4y
0yy
Now we use variation of parameters to find a particular solution:
261
4general
261
4
661
p
661594
933458
34
44
4p
tt
Cy
tt
ty
tvtvttv
tvt4vt4tvtt
v4
t
vt4tvt
t
v
t
4
t
v
t
vy
Quotient rule
Multiply both sides by t8
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
1c) (solve the initial value problem)
1)0(y;e2y3y t3
First we solve the homogeneous equation:
t3h Cey
0y3y
Now we use variation of parameters to find a particular solution:
t3t3general
t3p
t3t3t3t3
t3t3t3
t3p
et2Cey
et2y
t2v
2v
e2ev3ev3ev
e2ev3ev
evy
Now plug in the initial value to find C.
t3t3
0303
et2e)t(y
C1
e02Ce1
Solution to IVP
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
1d) (solve the initial value problem)
1)(y);tsin(y)ttan(y
First we solve the homogeneous equation:
)tsec(Cy
C)tcos(lnyln
dt)ttan(y
dyy)ttan(
0y)ttan(y
h
dtdy
Now we use variation of parameters to find a particular solution:
)tsec()t(sin)tsec(Cy
)tsec()t(siny
)t(sindt)tcos()tsin(v
)tcos()tsin(v)tsin()tsec(v
)tsin()tsec(v)ttan()ttan()tsec(v)tsec(v
)tsin()tsec(v)ttan()tsec(v
)tsec(vy
221
general
221
p
221
p
Now plug in the initial value to find C.
)tsec()t(sin)tsec()t(y
C1101C1
)sec()(sin)sec(C1
221
21
221
Solution to IVP
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
2) Farmer Fred opens a bank account with an initial deposit of $10,000. The account earns a monthly interest rate of 0.5%, compounded continuously. In addition, Fred will deposit $100 each month. How many months will it take for the account value to double?
Define variables: y(t) = account value at time t (months)
The differential equation is 100y005.0dt
dy We also have an initial value y(0)=10,000
We could solve this equation by separation, but let’s use the excellent guess method instead.
First we solve the homogeneous equation:
t005.0h Cey0y005.0y
For the particular solution, notice that we should get a constant. Plug in and solve:
000,20Cey
000,20y
000,20A100A005.00
t005.0general
p
005.0100
Now plug in the initial value to find C.
000,20e000,30)t(y
000,30C000,20Ce000,10t005.0
0005.0
Finally we can use our formula to find the answer. Just set y(t)=20,000 and solve for t.
months5.57005.0
)ln(t)ln(t005.0e
000,20000,20e000,30
34
34
34t005.0
t005.0
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
3) A lukewarm beverage (initially at 70°F) is placed into a cold (40°F) refrigerator. Five minutes later the temperature of the still-not-cold-enough beverage is 60°F. It is replaced in the fridge to continue the chilling process. How many minutes will it take for the beverage to reach the temperature of optimal refreshment (43°F)? Assume that the temperature follows Newton’s law of cooling, i.e. the rate of change of the temperature is proportional to the difference between the temperature of the beverage and the temperature of the surroundings.
Define variables: T(t)=temperature of beverage (in°F) at time t (minutes)
)40T(kdt
dTThe differential equation is Here k is a constant of proportionality
Let’s solve this one by separation.
40CeT
Ckt40Tln
dtk40T
dT)40T(k
dt
dT
kt
We need to find the 2 constants C and k. From the given information we know y(0)=70 and y(5)=60.
081.05
)ln(ke40e3060
30C40Ce70
32
32k5k5
0k
Now we can rewrite our formula with the constants we just found, then use it to solve the problem.
utesmin4.28081.0
)ln(t
e40e3043
40e30)t(T
101
101t081.0t081.0
t081.0
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
4) Consider the following autonomous differential equation:
02yyy 2
a) Sketch a slope field for this equation.b) Find any equilibrium solutions and classify them as stable or unstable.c) Find an explicit formula for the solution that passes through the initial value y(0) = 1.
a) The slope field shows the equilibrium solutions, and their stability. We will calculate them algebraically as well.
b) To find equilibrium solutions, set y’=0:
1y;2y
0)1y)(2y(
02yy2
Equilibrium solutions are the horizontal lines y=2 and y=-1
To assess the stability, plug in values on either side to find if the slope is positive or negative.
stable1y
unstable2y
4)2(y;2)0(y;2)3(y
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
4) Consider the following autonomous differential equation:
02yyy 2
c) Find an explicit formula for the solution that passes through the initial value y(0) = 1.
We will solve this by separation:
t3
31
2
Ce1y
2y
Ct31y
2yln
Ct31yln2yln
Ctdy1y
1
2y
1
dt)1y)(2y(
dy2yy
dt
dy
We use partial fractions to simplify the integral:
1y2y)1y)(2y(
1
B1y
A2y
)2y(B)1y(A1
1y
B
2y
A
)1y)(2y(
1
31
31
31
31
Use the initial value to find C
t321
2103
e1y
2y
CCe11
21
This is an implicit solution – we can
do some algebra and solve for y to get an explicit solution
Careful with the absolute values – in this case we have y(0)=1, so we can drop the abs value bars on the bottom, but we have to switch the top of the fraction.
t3
21
t321
t321t3
21
e1
e2y1yey2e
1y
y2
This solution is valid for -1<y<2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
5) Consider the following system of differential equations:
xy5.0y3dt
dy
xyx2dt
dx
a) Find and graph the nullclines of this system.b) What are the equilibria of this system?c) Classify any equilibria as stable or unstable.d) Sketch a reasonable solution to this system of differential equations.
Set x’=0 to find v-nullcline:
Set y’=0 to find h-nullcline:
0y;6x0xy5.0y3dt
dy
2y;0x0xyx2dt
dx
Equlibrium Points are
(0,0) – unstable
(6,2) - stable
h-nullclines in yellow
v-nullclines in red
Equilibria at intersections
Unstable equil. Stable equil.
Solution curve
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
6) Solve the following system of linear equations:
2z4yx3
5z2yx
1zyx
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
6) Solve the following system of linear equations:
2z4yx3
5z2yx
1zyx
Augmented matrix form
2
5
1
413
211
111
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
6) Solve the following system of linear equations:
2z4yx3
5z2yx
1zyx
Augmented matrix form
2
5
1
413
211
111
5
4
1
000
120
111
2R3R*3R
1
4
1
120
120
1111R2R*2R
1R33R*3R
Row reduction:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
6) Solve the following system of linear equations:
2z4yx3
5z2yx
1zyx
Augmented matrix form
2
5
1
413
211
111
5
4
1
000
120
111
2R3R*3R
1
4
1
120
120
1111R2R*2R
1R33R*3R
Row reduction:
The 3rd row represents the equation 0x+0y+0z = -5. This equation has no solution, so the original system of equations is inconsistent (not solvable).
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
7) Consider the following matrix:
4000
01300
4010
30226
A
a)Find the determinant of this matrix.b)Is this matrix invertible? If so, find the inverse.c)What is the determinant of the inverse of this matrix?d)Using this information, solve the following system of linear equations:
12w4
26z13
3yw4
2w3y22x6
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
7) Consider the following matrix:
4000
01300
4010
30226
A
a)Find the determinant of this matrix.b)Is this matrix invertible? If so, find the inverse.c)What is the determinant of the inverse of this matrix?d)Using this information, solve the following system of linear equations:
12w4
26z13
3yw4
2w3y22x6
a) Since A is a triangular matrix, the determinant is the product of the diagonal elements – det(A) = -312
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
7) Consider the following matrix:
4000
01300
4010
30226
A
a)Find the determinant of this matrix.b)Is this matrix invertible? If so, find the inverse.c)What is the determinant of the inverse of this matrix?d)Using this information, solve the following system of linear equations:
12w4
26z13
3yw4
2w3y22x6
a) Since A is a triangular matrix, the determinant is the product of the diagonal elements – det(A) = -312
b) Since det(A) is not zero, A is invertible, and we can use row reduction to find the inverse.
1A
41
131
2485
311
61
Identity
131*61*
41
485
*
41
43
*
*
41
41*
*
000
000
1010
0
1000
0100
0010
0001
3R3R
1R1R
000
0100
1010
0221
1000
01300
0010
0006
2R221R1R
000
0100
1010
001
1000
01300
0010
00226
4R42R2R
4R31R1R
000
0100
0010
0001
1000
01300
4010
30226
4R4R
2R2R
1000
0100
0010
0001
4000
01300
4010
30226
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
7) Consider the following matrix:
4000
01300
4010
30226
A
a)Find the determinant of this matrix.b)Is this matrix invertible? If so, find the inverse.c)What is the determinant of the inverse of this matrix?d)Using this information, solve the following system of linear equations:
12w4
26z13
3yw4
2w3y22x6
c) Since A-1is a triangular matrix, the determinant is the product of the diagonal elements: det(A -1) = -1/312
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
7) Consider the following matrix:
4000
01300
4010
30226
A
a)Find the determinant of this matrix.b)Is this matrix invertible? If so, find the inverse.c)What is the determinant of the inverse of this matrix?d)Using this information, solve the following system of linear equations:
12w4
26z13
3yw4
2w3y22x6
c) Since A-1is a triangular matrix, the determinant is the product of the diagonal elements: det(A -1) = -1/312
d) Notice that we have the inverse for the coefficient matrix, so we just need to multiply:
12
26
3
2
000
000
1010
0
w
z
y
x
41
131
2485
311
61
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
7) Consider the following matrix:
4000
01300
4010
30226
A
a)Find the determinant of this matrix.b)Is this matrix invertible? If so, find the inverse.c)What is the determinant of the inverse of this matrix?d)Using this information, solve the following system of linear equations:
12w4
26z13
3yw4
2w3y22x6
c) Since A-1is a triangular matrix, the determinant is the product of the diagonal elements: det(A -1) = -1/312
d) Notice that we have the inverse for the coefficient matrix, so we just need to multiply:
3
2
9
w
z
y
x
12
26
3
2
000
000
1010
0
w
z
y
x
6187
41
131
2485
311
61
8) Consider the following set of vectors in R3
3
2
1
a,
2
1
1
a,
1
1
0
a,
1
0
1
a 4321
a)What is the dimension of the span of this set of vectors?b)Is the span of this set of vectors R3? c)Is this a basis for R3?d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.
4
2
2
v,
3
0
2
v 21
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
8) Consider the following set of vectors in R3
3
2
1
a,
2
1
1
a,
1
1
0
a,
1
0
1
a 4321
4
2
2
v,
3
0
2
v 21
a) Put the column vectors in a matrix and perform elementary row operations:
0000
2110
1101
2R3R3R
2110
2110
1101
1R3R3R
3211
2110
1101**
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is the dimension of the span of this set of vectors?b)Is the span of this set of vectors R3? c)Is this a basis for R3?d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.
8) Consider the following set of vectors in R3
3
2
1
a,
2
1
1
a,
1
1
0
a,
1
0
1
a 4321
4
2
2
v,
3
0
2
v 21
a) Put the column vectors in a matrix and perform elementary row operations:
0000
2110
1101
2R3R3R
2110
2110
1101
1R3R3R
3211
2110
1101**
This has 2 pivot columns, so there are 2 linearly independent vectors in the set.
In other words, the span of this set is 2-dimensional.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is the dimension of the span of this set of vectors?b)Is the span of this set of vectors R3? c)Is this a basis for R3?d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.
8) Consider the following set of vectors in R3
3
2
1
a,
2
1
1
a,
1
1
0
a,
1
0
1
a 4321
4
2
2
v,
3
0
2
v 21
a) Put the column vectors in a matrix and perform elementary row operations:
0000
2110
1101
2R3R3R
2110
2110
1101
1R3R3R
3211
2110
1101**
This has 2 pivot columns, so there are 2 linearly independent vectors in the set.
In other words, the span of this set is 2-dimensional.
b) No, the span is 2-dimensional, and R3 is 3-dimensional.
c) No, this is not a basis because is does not span R3.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is the dimension of the span of this set of vectors?b)Is the span of this set of vectors R3? c)Is this a basis for R3?d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.
8) Consider the following set of vectors in R3
3
2
1
a,
2
1
1
a,
1
1
0
a,
1
0
1
a 4321
4
2
2
v,
3
0
2
v 21
a) Put the column vectors in a matrix and perform elementary row operations:
0000
2110
1101
2R3R3R
2110
2110
1101
1R3R3R
3211
2110
1101**
This has 2 pivot columns, so there are 2 linearly independent vectors in the set.
In other words, the span of this set is 2-dimensional.
b) No, the span is 2-dimensional, and R3 is 3-dimensional.
c) No, this is not a basis because is does not span R3.
d) We have several options for this part. One way is to make a good guess. If you notice that v2 is just 2 times a3 then we have our answer for that one:
2
1
1
2
4
2
2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is the dimension of the span of this set of vectors?b)Is the span of this set of vectors R3? c)Is this a basis for R3?d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.
8) Consider the following set of vectors in R3
3
2
1
a,
2
1
1
a,
1
1
0
a,
1
0
1
a 4321
4
2
2
v,
3
0
2
v 21
a) Put the column vectors in a matrix and perform elementary row operations:
0000
2110
1101
2R3R3R
2110
2110
1101
1R3R3R
3211
2110
1101**
This has 2 pivot columns, so there are 2 linearly independent vectors in the set.
In other words, the span of this set is 2-dimensional.
b) No, the span is 2-dimensional, and R3 is 3-dimensional.
c) No, this is not a basis because is does not span R3.
d) We have several options for this part. One way is to make a good guess. If you notice that v2 is just 2 times a3 then we have our answer for that one:
2
1
1
2
4
2
2
For v1 there is no obvious answer, so we need to try to solve for it. Since we know that our span is only 2-dimensional, we only need two basis vectors. Choose any 2 independent vectors from our set, say a1 and a2, and try to write v1 as a linear combination of them.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is the dimension of the span of this set of vectors?b)Is the span of this set of vectors R3? c)Is this a basis for R3?d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.
8) Consider the following set of vectors in R3
3
2
1
a,
2
1
1
a,
1
1
0
a,
1
0
1
a 4321
4
2
2
v,
3
0
2
v 21
a) Put the column vectors in a matrix and perform elementary row operations:
0000
2110
1101
2R3R3R
2110
2110
1101
1R3R3R
3211
2110
1101**
This has 2 pivot columns, so there are 2 linearly independent vectors in the set.
In other words, the span of this set is 2-dimensional.
b) No, the span is 2-dimensional, and R3 is 3-dimensional.
c) No, this is not a basis because is does not span R3.
d) We have several options for this part. One way is to make a good guess. If you notice that v2 is just 2 times a3 then we have our answer for that one:
2
1
1
2
4
2
2
For v1 there is no obvious answer, so we need to try to solve for it. Since we know that our span is only 2-dimensional, we only need two basis vectors. Choose any 2 independent vectors from our set, say a1 and a2, and try to write v1 as a linear combination of them.
ntInconsiste
cc3
c0
c2
c
c
0
c
0
c
3
0
2
acacv
21
2
1
2
2
1
1
22111
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is the dimension of the span of this set of vectors?b)Is the span of this set of vectors R3? c)Is this a basis for R3?d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.
8) Consider the following set of vectors in R3
3
2
1
a,
2
1
1
a,
1
1
0
a,
1
0
1
a 4321
4
2
2
v,
3
0
2
v 21
a) Put the column vectors in a matrix and perform elementary row operations:
0000
2110
1101
2R3R3R
2110
2110
1101
1R3R3R
3211
2110
1101**
This has 2 pivot columns, so there are 2 linearly independent vectors in the set.
In other words, the span of this set is 2-dimensional.
b) No, the span is 2-dimensional, and R3 is 3-dimensional.
c) No, this is not a basis because is does not span R3.
d) We have several options for this part. One way is to make a good guess. If you notice that v2 is just 2 times a3 then we have our answer for that one:
2
1
1
2
4
2
2
For v1 there is no obvious answer, so we need to try to solve for it. Since we know that our span is only 2-dimensional, we only need two basis vectors. Choose any 2 independent vectors from our set, say a1 and a2, and try to write v1 as a linear combination of them.
ntInconsiste
cc3
c0
c2
c
c
0
c
0
c
3
0
2
acacv
21
2
1
2
2
1
1
22111
We get an inconsistent set of equations, so v1 is not in the span. Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is the dimension of the span of this set of vectors?b)Is the span of this set of vectors R3? c)Is this a basis for R3?d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.
9) Consider the following set of vectors in P2:
x2a,xx1a,xx1a 32
22
1
a)What is dimension of the span of this set?b)Does this set span P2?
c)Is this set a basis for P2?
d)Express each of the standard basis vectors {1, x, x2} as a linear combination of the vectors in the set.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
9) Consider the following set of vectors in P2:
x2a,xx1a,xx1a 32
22
1
a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is dimension of the span of this set?b)Does this set span P2?
c)Is this set a basis for P2?
d)Express each of the standard basis vectors {1, x, x2} as a linear combination of the vectors in the set.
9) Consider the following set of vectors in P2:
x2a,xx1a,xx1a 32
22
1
a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out:
tindependen0ccc
0c2cc
0ccc
0cc
?0)c2cc(x)ccc(x)cc(
?0)x2(c)xx1(c)xx1(c
?0acacac
321
321
321
21
3213212
21
32
22
1
332211
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is dimension of the span of this set?b)Does this set span P2?
c)Is this set a basis for P2?
d)Express each of the standard basis vectors {1, x, x2} as a linear combination of the vectors in the set.
9) Consider the following set of vectors in P2:
x2a,xx1a,xx1a 32
22
1
a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out:
tindependen0ccc
0c2cc
0ccc
0cc
?0)c2cc(x)ccc(x)cc(
?0)x2(c)xx1(c)xx1(c
?0acacac
321
321
321
21
3213212
21
32
22
1
332211
b) P2 is a 3-dimensional space, and we have 3 independent vectors in P2 – yes they span P2.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is dimension of the span of this set?b)Does this set span P2?
c)Is this set a basis for P2?
d)Express each of the standard basis vectors {1, x, x2} as a linear combination of the vectors in the set.
9) Consider the following set of vectors in P2:
x2a,xx1a,xx1a 32
22
1
a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out:
tindependen0ccc
0c2cc
0ccc
0cc
?0)c2cc(x)ccc(x)cc(
?0)x2(c)xx1(c)xx1(c
?0acacac
321
321
321
21
3213212
21
32
22
1
332211
b) P2 is a 3-dimensional space, and we have 3 independent vectors in P2 – yes they span P2.
c) P2 is a 3-dimensional space, and we have 3 independent vectors in P2 – yes they form a basis for P2.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is dimension of the span of this set?b)Does this set span P2?
c)Is this set a basis for P2?
d)Express each of the standard basis vectors {1, x, x2} as a linear combination of the vectors in the set.
9) Consider the following set of vectors in P2:
x2a,xx1a,xx1a 32
22
1
a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out:
tindependen0ccc
0c2cc
0ccc
0cc
?0)c2cc(x)ccc(x)cc(
?0)x2(c)xx1(c)xx1(c
?0acacac
321
321
321
21
3213212
21
32
22
1
332211
b) P2 is a 3-dimensional space, and we have 3 independent vectors in P2 – yes they span P2.
c) P2 is a 3-dimensional space, and we have 3 independent vectors in P2 – yes they form a basis for P2.
d) Form a linear combination of a1, a2 and a3:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is dimension of the span of this set?b)Does this set span P2?
c)Is this set a basis for P2?
d)Express each of the standard basis vectors {1, x, x2} as a linear combination of the vectors in the set.
9) Consider the following set of vectors in P2:
x2a,xx1a,xx1a 32
22
1
a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out:
tindependen0ccc
0c2cc
0ccc
0cc
?0)c2cc(x)ccc(x)cc(
?0)x2(c)xx1(c)xx1(c
?0acacac
321
321
321
21
3213212
21
32
22
1
332211
b) P2 is a 3-dimensional space, and we have 3 independent vectors in P2 – yes they span P2.
c) P2 is a 3-dimensional space, and we have 3 independent vectors in P2 – yes they form a basis for P2.
d) Form a linear combination of a1, a2 and a3:
3221
121
321
221
1
321
321
21
2321321
221332211
aaa1
1c;c;c
1c2cc
0ccc
0cc
1x0x0)c2cc(x)ccc(x)cc(1acacac
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is dimension of the span of this set?b)Does this set span P2?
c)Is this set a basis for P2?
d)Express each of the standard basis vectors {1, x, x2} as a linear combination of the vectors in the set.
9) Consider the following set of vectors in P2:
x2a,xx1a,xx1a 32
22
1
a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out:
tindependen0ccc
0c2cc
0ccc
0cc
?0)c2cc(x)ccc(x)cc(
?0)x2(c)xx1(c)xx1(c
?0acacac
321
321
321
21
3213212
21
32
22
1
332211
b) P2 is a 3-dimensional space, and we have 3 independent vectors in P2 – yes they span P2.
c) P2 is a 3-dimensional space, and we have 3 independent vectors in P2 – yes they form a basis for P2.
d) Form a linear combination of a1, a2 and a3:
3221
121
321
221
1
321
321
21
2321321
221332211
aaa1
1c;c;c
1c2cc
0ccc
0cc
1x0x0)c2cc(x)ccc(x)cc(1acacac
Similar analysis yields the other 2 combinations.
3221
1212
321
a0aax
aaax
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
a)What is dimension of the span of this set?b)Does this set span P2?
c)Is this set a basis for P2?
d)Express each of the standard basis vectors {1, x, x2} as a linear combination of the vectors in the set.
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