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Unformatted text preview: Math 136 Assignment 3 Due: Wednesday, Jan 27th 1. For each of the following systems of linear equations:
i) Write the augmented matrix.
ii) Row—reduce the augmented matrix into row echelon form. iii) Find the general solution of the system or explain why the system is inconsistent. a) 1‘1 '1' 332 = “‘7
2x1 + 4:52 + $3 = —16
1‘1 + 2.712 + 1‘3 : 9
b) $1 + $2 + 2.123 + m4 = 3
:51 + 2352 + 4353 + m4 = 7
(131 + 334  = “21
c) (1 + i)z1 + 2222 2 1
. 1 l .
(1 + Z)22 + 23 = 5 — 52
21 “‘ Z3 = 0 2. Determine if the vector 1; is in the span of the given set S. If it is, write I; as
a linear combination of the vectors in S. a) s : {(2,’3, 1), (1,5,0), (1,—2, 1), (4, —6, —2
b) s = {(1, 2, 1, 1), (2, —3,3,4), (3, —6,4,5)}, \_/ }, 13: (6,8, —1).
‘ (1,—1,1,1). on
H 3. What can you say about the consistency and the number of parameters in the general
solution of a system of m linear equations in 71 variables if: a) m = 5, n = 7, the rank of the coefﬁcient matrix is 4.
b) m = 3, n = 6, the rank of the coefﬁcient matrix is 3. c) m = 5, n = 4, the rank of the augmented matrix is 4. 4. Let S : {551, . . . ,fk} be a set of k vectors in R".
a) Prove that if span S = R", then k 2 n. b) Consider the following statement “If k 2 n, then span 5’ = R" ”.
Give a proof if the statement is true or a counter example if the statement is false. Use MATLAB to complete the following questions. You do not need to submit a printout of your work. Simply use MATLAB to solve the problems,
and submit written answers to the questions along with the rest of your assignment. Applications of Systems of Equations A problem in the study of heat transfer is to determine the steady—state temperature distribution
of a thin plate when the temperature around the boundary is known. Consider the square plate shown below: 37° 37° 37° 42° 42° 42° Let T1, . . . ,Tg denote the temperatures at the nine interior nodes shown above. When the plate
is in thermal equilibrium and the nodes are equally spaced, the temperature, Ti, at the ith
interior node is approximately given by the average of the temperatures at the four adjacent
points. For example, _ 37°+63°+T2+T6 T1 4 01' 4T1~T2—T6:100. The approximation gets better as the spacing between nodes decreases. ————I——_————u—_——_—__ Given the square plate and temperatures from the previous page: (a) Determine (on paper) the system of equations for the unknown temperatures T1, . . . ,Tg. (b) Enter the augmented matrix that represents this system in MATLAB and use MATLAB to
ﬁnd the temperatures. ' (c) As previously mentioned, the approximation gets better as the spacing between nodes
decreases. Consider the point, labelled by T10: 18° 18° Using the computed temperatures, T1, T2, T5, and T6, that you found in part (b), ﬁnd T10. ...
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