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MATH 511: Linear Algebra
Exam 5 Key
1. (40 pts.) Find the eigenvalues and the eigenspace corresponding to each
eigenvalue for each of the following matrices.
a)
3 4
1 3
Let A be the above matrix. To find the eigenvalues of A, we find the determinant
3
4
MATH 511: Linear Algebra
Homework 14 Key
Section 4.1, p. 182: 6, 9, 19
6. Determine whether the following are linear transformations from R2 into R3 .
a) L(
x) = (x1 , x2 , 1)T
To determine whether or not this is a linear transformation, we need to determ
MATH 511: Linear Algebra
Homework 15 Key
Section 4.2, p. 196: 3, 18
3. For each of the following linear operators L on R3 , find a matrix A such that
L(
x) = A
x for every x in R3 .
a) L(x1 , x2 , x3 )T ) = (x3 , x2 , x1 )T
To find the matrix representati
MATH 511: Linear Algebra
Homework 19 Key
Section 5.3, p. 243: 3, 5
3. For each of the following systems A
x = b, find all least squares solutions.
1
2
4 ,
a) A = 2
1 2
3
b = 2 .
1
We find the least squares solutions by means of the normal equations
AT A
x
MATH 511: Linear Algebra
Exam 4 Key
1. (15 pts.) Find the distance from the point (2, 1, 3) to the plane 3x2y +z = 0.
This is given by the absolute value of the scalar projection of the vector (2, 1, 3)T onto
= (3, 2, 1)T :
the vector N
h(2, 1, 3)T , (3
MATH 511: Linear Algebra
Homework 20 Key
Section 5.4, p. 252: 7, 24
7. In C[0, 1], with inner product defined by (3), compute:
a) hex , ex i
x
1
Z
x
ex ex dx
he , e i =
Z0 1
dx
=
0
= x|10
= 10
= 1.
b) hx, sin xi
Z
hx, sin(x)i =
1
x sin(x)dx
0
1
1
1
=
x c
MATH 511: Linear Algebra
Homework 16 Key
Section 4.3, p. 204: 5, 12
5. Let L be the operator on P3 defined by
L(p(x) = xp0 (x) + p00 (x).
a) Find the matrix A representing L with respect to [1, x, x2 ].
We find each of L(1), L(x), and L(x2 ) in terms of [
MATH 511: Linear Algebra
Homework 23 Key
Section 6.1
1.
Find the eigenvalues and the corresponding eigenspaces for each of the
following matrices.
a)
6 4
3 1
Let A be the above matrix. To find the eigenvalues of A, we find the determinant
6
4
det(A I) =
MATH 511: Linear Algebra
Exam 1
1. (40 pts.) Let the matrix A be given by
1
0
1
3
4 .
A= 3
2 2 3
a) Find the inverse of A.
We set up a matrix so that we can find the inverse.
1
0
1 1 0
3
3
4 0 1
2 2 3 0 0
Consider
0
0 .
1
We reduce this until the left bl
MATH 511: Linear Algebra
Homework 22 Key
Section 5.6, p. 281: 3, 8
3.
Given the basis cfw_(1, 2, 2)T , (4, 3, 2)T , (1, 2, 1)T for R3 , use the Gram-Schmidt
process to obtain an orthonormal basis.
Let x1 = (1, 2, 2)T , x2 = (4, 3, 2)T , and x3 = (1, 2, 1
MATH 511: Linear Algebra
Homework 24 Key
Section 6.2
1. Solve each of the following initial value problems.
a)
y10 = y1 2y2
y20 = 2y1 + y2
y1 (0) = 1, y2 (0) = 2
We can express this as
y10
y20
=
Let
1 2
2
1
A=
1 2
2
1
y1
y2
.
.
The eigenvalues of A are th
MATH 511: Linear Algebra
Homework 24 Key
Section 6.2, p. 323: 2ad, 6
2. Solve each of the following initial value problems.
a)
y10 = y1 + 2y2
y20 = 2y1 y2
y1 (0) = 3, y2 (0) = 1
We can express this as
y10
y20
=
Let
1
2
2 1
A=
1
2
2 1
y1
y2
.
.
The eigenva
MATH 511: Linear Algebra
Homework 18 Key
Section 5.2, p. 233: 1bc, 5
1. For each of the following matrices, determine a basis for each of the subspaces
R(AT ), N (A), R(A), and N (AT ).
b) A =
1 3 1
2 4 0
To find a basis for R(AT ) and N (A), we rewrite A
2/5/2017
Ch.3 PressureMeasurement
PROF.TAHAALDOSS
TEXTCH.6
Application:Barometer
A barometer measures local absolute atmospheric
pressure: the height of a mercury column is
proportional to patm;
AtmosphericPressure=theweightofairfromapoint
ontheearthsurfa
Computer Applications, ME 325, 2015 Fall
Computer Applications (ME 325), Hw #2
Due date and time: 9/2, 12:30 pm (before Class Time)
Note: PLEASE provide the detailed step(s), otherwise zero credit will be given.
1. Please determine if the following equati
Computer Applications, ME 325, 2015 Fall
Computer Applications (ME 325), Hw #6
Due date and time: 9/30, 12:30 pm (before Class Time)
Note: PLEASE provide the KEY STEP(S) to reach your solution, otherwise zero credit will
be given.
1. Consider the cross se
Computer Applications, ME 325, 2015 Fall
Computer Applications (ME 325), Hw #1
Due date and time: 8/26, 12:30 pm (before Class Time)
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1. (Reading assignment) Please CAREFULLY read
Computer Applications, ME 325, 2015 Fall
Computer Applications (ME 325), Hw #5
Due date and time: 9/23, 12:30 pm (before Class Time)
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be given.
1. Matt needs to write t
Computer Applications, ME 325, 2015 Fall
Computer Applications (ME 325), Hw #7
Due date and time: 10/7, 12:30 pm (before Class Time)
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be given.
1. (Differentiation) A f
Computer Applications, ME 325, 2015 Fall
Computer Applications (ME 325), Hw #3
Due date and time: 9/9, 12:30 pm (before Class Time)
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be given.
1. Consider the cross sec
Computer Applications, ME 325, 2015 Fall
Computer Applications (ME 325), Hw #4
Due date and time: 9/16, 12:30 pm (before Class Time)
Note: PLEASE provide the KEY STEP(S) to reach your solution, otherwise zero credit will
be given.
1. Please consider the l
MATH 511: Linear Algebra
Homework 25 Key
Section 6.3, p. 340: 1ace, 28ad
1. In each of the following, factor the matrix A into a product XDX 1 , where
D is diagonal.
a) A =
0 1
1 0
To do this, we must find the eigenvalues of A. These are given by 1 = 1 an
MATH 511: Linear Algebra
Exam 3
1. (30 pts.) Let E and F be the ordered bases in R3 given by
1
2
1
1
1
1
5 ,
3 ,
1
2 ,
1 ,
2 .
E=
, F =
4
2
1
1
0
0
a) Find the transition matrix from E to the standard basis [
e1 , e2 , e3 ].
This matrix is given by th
MATH 511: Linear Algebra
Homework 23 Key
Section 6.1, p. 310: 1cfil, 8
1.
Find the eigenvalues and the corresponding eigenspaces for each of the
following matrices.
c)
3 1
1
1
Let A be the above matrix. To find the eigenvalues of A, we find the determinan
MATH 511: Linear Algebra
Homework 21 Key
Section 5.5, p. 270: 1, 6, 27ab
1. Which of the following sets of vectors form an orthonormal basis for R2 ?
a) (1, 0)T , (0, 1)T
This set forms a basis for R2 , since R2 has dimension 2, and we have two linearly
MATH 511: Linear Algebra
Homework 17 Key
Section 5.1, p. 223: 3, 8
3. For each of the following pairs of vectors x and y, find the vector projection
p of x onto y and verify that p and x p are orthogonal.
a) x = (3, 4)T and y = (1, 0)T .
The vector projec