MAT1163 Linear Algebra Assignment 1, Semester 2 2014
Culbert
444.xlsxSheet1
| 1 | MAT1163 Linear Algebra | txt | ||||||
| txt | Assignment 1, Semester 2 2014 | |||||||
| Due: 9 am Monday 25th August 2014 | ||||||||
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| ~ | ||||||||
| (i) Pivot Columns | FALSE | FALSE | FALSE | FALSE | FALSE | |||
| (ii) Basic Variables | FALSE | FALSE | FALSE | FALSE | FALSE | |||
| (iii) Free Variables | FALSE | FALSE | FALSE | FALSE | FALSE | |||
| (iv) | ||||||||
| = | (Eq 1) | |||||||
| = | (Eq 2) | |||||||
| = | (Eq 3) | |||||||
| = | (Eq 4) | |||||||
| (v) General Solution | ||||||||
| x1 = | ||||||||
| x2 = | ||||||||
| x3 = | ||||||||
| x4 = | ||||||||
| x5 = | ||||||||
| IMPORTANT: Use the lower case letters r, s, t and u (as many as required) to denote the parameters in part v) above and vi), vii) below. | ||||||||
| vi) General solution in parametric vector form: | ||||||||
| x = | + r | + s | + t | |||||
| vii) General solution of homogeneous system. | ||||||||
| x = | r | + s | + t | |||||
| a = | ||||||||
| a = | ||||||||
| a = | Consistent | |||||||
| Inconsistent | ||||||||
| 1.00 | ||||||||
| x1 = | ||||||||
| x2 = | ||||||||
| x3 = | ||||||||
| IMPORTANT: Leave the answer cells for x1, x2, and x3 blank if you believe the system to be inconsistent. | ||||||||
| = | ||||||||
| h = | ||||||||
| 1 | ||||||||
| 1 | ||||||||
| 1. There are more vectors than the number of components per vector. | ||||||||
| 2. The 0 vector is in the set. | ||||||||
| 3. The vectors are not multiples of one another. | 1 | |||||||
| 4. The matrix whose columns are the given vectors has a pivot element in every column. | ||||||||
| 5. There are fewer columns than the number of components per vector. | 1 | |||||||
| 6. The reduced row echelon form of the matrix whose columns are the given vectors is the identity matrix. | ||||||||
| 7. The reduced row echelon form of the matrix whose columns are the given vectors is not equal to the identity matrix. | ||||||||
| 8. One of the vectors is a linear combination of the other vectors in the set. | 1 | |||||||
| 9. The vectors are multiples of one another. | ||||||||
| 10. The matrix whose columns are the given vectors does not have a pivot element in every column. | ||||||||
| 1.00 | ||||||||
| Linearly Independent | ||||||||
| Linearly Dependent | ||||||||
| 1 | ||||||||
| 1 | ||||||||
| 1 | ||||||||
| 1 | ||||||||
| 1 | ||||||||
| 1 | ||||||||
| T1 = | ||||||||
| T2 = | ||||||||
| T3 = | ||||||||
| T4 = | ||||||||
| END OF ASSIGNMENT | 1 | |||||||
Column 1
Column 2
Column 3
Column 4
Column 5
x1
x2
x3
x4
x5
x1
x2
x3
x4
x5
Sheet2
Sheet3
IMPORTANT: In what follows denote the variables
54321
and ,,, xxxxx
by x1, x2,
x3, x4 and x5.
(d) Use your result from (c) to determine
(i) Pivot Columns:
(ii) Basic variables:
(iii) Free variables
(iv) System of linear equations corresponding to the reduced row echelon for of
the augmented matrix:
(v) General solution:
(vi) Write the general solution in parametric vector form.
(vii) Determine the general solution of the corresponding homogeneous system.
Question 2
Consider the system of linear equations
1
1
8
4
7
6
3
2
1
3
2
1
3
2
1
=
+
+
=
+
+
=
+
+
ax
ax
ax
x
ax
ax
x
x
ax
where a is a constant.
a) For which 2 values of a will the augmented matrix have 3 pivots, but no solution for the linear system? For each value writ down an associated row echelon form to support your answer.
_1467715572.unknown
Question 2
Consider the system of linear equations
1
18
476
321
321
321
axaxax
xaxax
xxax
where a is a constant.
a) For which 2 values of a will the augmented matrix have 3 pivots, but no solution for
the linear system? For each value writ down an associated row echelon form to
support your answer.
Question 3
a) Find a condition that h and k need to satisfy so that the set of vectors
ï
þ
ï
ý
ü
ï
î
ï
í
ì
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
k
h
1
,
4
6
2
,
4
2
1
is linearly dependent. Terms involving h and k are to appear on the left of the equals sign below, while any constant term is to appear on the right.
_1467718307.unknown
Question 3
a) Find a condition that h and k need to satisfy so that the set of vectors
k
h
1
,
4
6
2
,
4
2
1
is linearly dependent. Terms involving h and k are to appear on the left of the equals
sign below, while any constant term is to appear on the right.
b) For which value of a will the augmented matrix have fewer than 3 pivots? Is the corresponding system of equations consistent? If yes, determine the solution of the system for this value of a.
b) For which value of a will the augmented matrix have fewer than 3 pivots? Is the
corresponding system of equations consistent? If yes, determine the solution of the
system for this value of a.
b) Determine the value of h if k = 4 and write down a non-trivial dependence relation for the resulting three vectors.
b) Determine the value of h if k = 4 and write down a non-trivial dependence relation
for the resulting three vectors.
+
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
4
2
1
4
2
1
+
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
4
6
2
4
6
2
ú
ú
ú
û
ù
ê
ê
ê
ë
é
=
ú
ú
ú
û
ù
ê
ê
ê
ë
é
0
0
0
4
1
h
0
0
0
4
1
h
Question 4
For each of the following vectors decide if they are linearly independent. In each case give a reason to support your answer
Question 4
For each of the following vectors decide if they are linearly independent. In each case
give a reason to support your answer
(a)
ú
û
ù
ê
ë
é
ú
û
ù
ê
ë
é
ú
û
ù
ê
ë
é
0
1
,
1
2
,
3
1
(b)
ú
û
ù
ê
ë
é
-
ú
û
ù
ê
ë
é
6
3
,
2
1
(c)
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é
4
0
0
,
2
1
2
,
2
1
1
(d)
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é
3
1
4
,
0
2
2
,
2
0
2
(e)
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
-
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é
6
0
3
,
10
4
7
,
8
2
5
_1467719927.unknown
_1467719984.unknown
_1467720063.unknown
_1467719952.unknown
_1467719892.unknown
(a)
0
1
,
1
2
,
3
1
(b)
6
3
,
2
1
(c)
4
0
0
,
2
1
2
,
2
1
1
(d)
3
1
4
,
0
2
2
,
2
0
2
(e)
6
0
3
,
10
4
7
,
8
2
5
(f)
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
-
-
0
0
0
1
,
0
0
1
1
,
1
1
0
1
,
2
2
2
2
_1467720197.unknown
Question 1
Consider the system of linear equations
6
14
4
11
2
18
22
2
20
2
2
12
23
6
18
3
6
4
2
3
5
4
3
2
1
5
4
3
2
1
5
4
3
2
1
5
4
3
2
1
=
-
-
+
-
-
=
-
-
+
-
=
-
-
+
-
-
=
+
+
+
+
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
.(a) Write down the augmented matrix corresponding to this system.
_1467710363.unknown
(f)
0
0
0
1
,
0
0
1
1
,
1
1
0
1
,
2
2
2
2
Question 5
We wish to determine the steady state temperature distribution of a thin plate when the temperature around the boundary is known. Assume that the plate shown below represents a cross section of a metal beam with negligible heat flow in the direction perpendicular to the plate. The temperatures at the four interior mesh nodes shown in the figure are denoted by
4
T
. The temperature at a node is approximately equal to the average of the four nearest nodes, i.e. to the left, right, above and below.
(a) Write a system of four equations whose solution gives estimates for the temperatures
3
2
1
,
,
T
T
T
and4
T
. Enter the equation coefficients in the augmented matrix below.3
2
1
,
,
T
T
T
_1464370797.unknown
_1464370859.unknown
Question 5
We wish to determine the steady state temperature distribution of a thin plate when the
temperature around the boundary is known. Assume that the plate shown below
represents a cross section of a metal beam with negligible heat flow in the dire ction
perpendicular to the plate. The temperatures at the four interior mesh nodes shown in
the figure are denoted by
321
,,TTT
and
4
T
. The temperature at a node is approximately
equal to the average of the four nearest n odes, i.e. to the left, right, above and below.
(a) Write a system of four equations whose solution gives estimates for the
temperatures
321
,,TTT
and
4
T
. Enter the equation coefficients in the
augmented matrix below.
(b) Solve the system you derived in (a) for
3
2
1
,
,
T
T
T
and4
T
._1464370797.unknown
_1464370859.unknown
(b) Solve the system you derived in (a) for
321
,,TTT
and
4
T
.
Question 1
Consider the system of linear equations
6144112
182222022
12236183
6423
54321
54321
54321
54321
xxxxx
xxxxx
xxxxx
xxxxx
.
(a) Write down the augmented matrix corresponding to this system.
We use elementary row operations to determine the solution of the system.
(b) Use the drop-down menu’s below to write down a sequence of elementary row operations to determine a row echelon matrix that is row-equivalent to the augmented matrix. Write down the matrix you obtained.
We use elementary row operations to determine the solution of the system.
(b) Use the drop-down menu’s below to write down a sequence of elementary row
operations to determine a row echelon matrix that is row-equivalent to the
augmented matrix. Write down th e matrix you obtained.
(c) Identify a sequence of elementary row operations that will reduce the matrix you found in (b) to reduced row echelon form and hence write the down the reduced row echelon form for the augmented matrix you determined in (a).
(c) Identify a sequence of elementary row operations that will reduce the matrix you
found in (b) to reduced row echelon form and hence write the down the reduced row
echelon form for the augmented matrix you determined in (a).
IMPORTANT: In what follows denote the variables
5
4
3
2
1
and
,
,
,
x
x
x
x
x
by x1, x2, x3, x4 and x5.(d) Use your result from (c) to determine
(i) Pivot Columns:
(ii) Basic variables:
(iii) Free variables
(iv) System of linear equations corresponding to the reduced row echelon for of the augmented matrix:
(v) General solution:
(vi) Write the general solution in parametric vector form.
(vii) Determine the general solution of the corresponding homogeneous system.
