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This course includes introduction and operation on matrices, Cramer rule, eigen values and eigen vector, complex matrices, LU factorization, linear system, vector operations, least square curves. This file have solved problems set. It includes: Linear, Transformation, Region, Mapping, Span, Vector, Basis, Row, Echleon, Form, Augmented, Matrix
Typology: Exercises
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Question: Find row echelon form of the matrix
Solution:
34 11 43
32
4 3 4 (( 11 // 33 ))
B
C D 100
200
100
200
Thus the row echelon form of the given matrix is
Question: The flow of traffic at the kalma chowk on Ferozepur road ,Lahore is shown below. Construct system of linear equations representing flow using the principle “In flow at a point is equal to outflow at that point.”
(i): Discuss the consistency of the system. (ii): Find two possible flows. Solution: Node A: Node B: 1 Node C: Node D:
x (^1) A x^2
x (^4) x 3
Node C: Node D: Augmented matrix is
Row echelon form is
System is consistent and has infinite solutions as Rank of Augmented matrix = Rank of coefficient matrix
Two possible flows = Two solutions of the system. These solutions are for
. i.e., [ ] [ ]
Question: The following sets are not vector spaces. Indicate all condition that fail to hold with explaination. i. The set under the operations ( ) ( ) (^) ( ) ii. The set under the operations^ (^ )^ (^ )
( ) (^) ( ( (^) ) () ( (^) ) ) Solution: (i). The following conditions fail.
eigenvector is [ ] √ eigenvector is [(( √√^ ))^ ((^ √√^ ))] √ eigenvector is [(( √√^ ))^ ((^ √√^ ))]
Question no. 3: Find conditions on *( ) ( ) ( so that)+. ( ) in belongs to ( ), where Solution: For ( ) to be in W , the solution of the system ( ) ( ) ( ) ( ) must exist. The augmented matrix is (^) 221 123 10117 cba whose row echelon form is
Question no. 4: Find a basis and nullity for the null space of the matrix
Answer:
{[ ] [ ]}
