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This is solution to assignment of Basic Mathematics course. This was submitted to Karunashankar Sidhu at Institute of Mathematical Sciences. It includes: Images, Transformations, Maps, Matrices, Linear, Equations, Solution, Elementary, Row, Operation
Typology: Exercises
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Maximum Marks: 30 Due Date: April 25, 2012
Question: 1 Marks: 10
Let (^1 2 1 )
e e y and y
Let T: R^2 R^2 be a linear transformation that maps e (^1)
into y 1 and maps e 2 into y 2. Find the images of 1 2
x and x
Solution:
1 2 1 2
1 2
1 2
1 2 1 2 1 1 2 2
e e e and e
e e
So
T T e e
T e T e y y T e y and T e y
Therfore the image of is
1 1 1 2 2 2 1 1 1 2 2 2 1 1 2 2
1 2
1 2 1 2 1 1 2 2 1 2
Similarly x x e x e x x T T x e x e x x T e x T e
x x
x x x x x x x Therfore the image of is x x x
Question: 2 Marks: 10
Using elementary row operations, find the inverse of the following matrix, if it exists. 3 4 1 1 0 3 2 5 4
Solution:
Let
det 1 0 3 3( 15) 1( 11) 2(12) 45 11 24 10 2 5 4
As the given matrix is non-singular, therefore, inverse of the matrix is possible. We reduce it to reduced echelon form.
1 3 2 3
Hence the inverse of the original matrix is 1
Question: 3 Marks: 5
Find an LU factorization of the matrix
Solution: We will reduce A to a row echelon form U and at each step we will fill in an entry of L in accordance with the four-step procedure. 3 6 3 6 7 2 1 7 0
multiplier
multiplier multiplier
multiplier
multiplier
U^ multiplier
So
3 0 0 1 2 1 6 5 0 0 1 4 5 (^1 5 5 0 0 )