Download DU (University of Delhi) B-Tech 2nd Semester 7801 Linear Algebra for Comp. Sc. Question Paper

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[This question paper cuntains 4 printed pages.)
Sr.No. onucstion Paper : 7801 F?2 ~. Your Roll No
Unique Paper Code : 234120]

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Name of the Course : B.Tech. in Computer Science
v N-alnc of the Paper : Linear Algebra for Comp. Sc.
Semester : II
Duration : 3 Hours Maximum Marks : 75
" Instructions for Candidates

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1. Write your Roll No. on the top immediately on receipt of this question paper.
2 Attempt five questions in all.
3. Question No. 1 is compulsory.
4
All the symbols have their usual meanings.

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l. (a) F ind the inverse of the following matrix using elementary row transfonnations
' 3?1-2
v 20?1
_ ? 3?so\
' 1 2 2 4

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V (b) Describe the null space of the matrix A = [ :l.
3 8 6 16
v . (c) State anctprovc Cauchy-Schwarz?s inequality.
(d) Use Cramer?s rule to solve Ax = b, where.
l 0 l

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2 1 and b = 0
l 2 0
R730.

v \?K

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' 780-1 2 2K
(6) Find the Eigen values and the corresponding Eigen vectors ofthe matrix \
v 1 1 0 \
v A = 0 l 0 \
0 0 l '

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v (0 Let M be the space of all n X n matrices. Let T: M ?> M be a map such that
l
_ T(A) = A_TA . Show that T is linear. Also show that the kernel of T consists
' of the space of symmetric matrices.
' (g) The fraction of rental cars in a city starts at 0.02. The fraction outside

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' that city is 0.98. Every month, 80% of the cars stay in the city (and 20%
- leave). Also 5% of the outside cars come in the city (95% stay outside).
., Estimate the fraction of cars coming and leaving the city at the end of 5
_ ' months. V? (5x7=35)
- r; '

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2. (a) F ind the complete solution to Ax = b, where
v 1 3 0 2 1
_ A = 0 0 I 4 and b = 6
1 3 1 6 7
- . ' . . .\

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(b) Wnte the LU-decomposxtlon of the matnx
3 ?6 -3
v
A = 2 0 6
' ?4 7 5

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- Also compute 33,153: and E; to find L. (5,5)
3. (8) De?ne column space and null space of a matrix A. Let Ax = b be the system
of m linear equations in n unknowns. Prove that the column space and null
' space of A are the subspaces of R" and Rll respectively.
v

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v v v v v '-
v
780] 3
v .

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_ (b) Find the bases and dimensions for all four fundamental subspaces 0fthe
following matrix A, namely, the column space ofA, the null space of A, the
v t
1 row spaee ofA and the null space of AT. Ajso determine the rank of the
' matrix A.

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v
" 1 3 1 2
' A = 2 6 4 8 (5,5)
- O 0 2 4
v

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V 4.
(3) Use Gram-Schmidt?s orthogonalisation process to ?nd the QR decomposition
- ( of the matrix
v " 102
v (b) Find the pivots of the given matrix A and verify that det.A = product of the

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pivots.
(5.5)
G
>
Il

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uro?
uNN
uuw
/
- 5. (a) Diagonalise the following matrix A and hence ?nd A?.

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v [3 4] t
A =
- 4 ?3 r
? ? 2 2
(b) Find the singular value decomposition (svd) of the matrix A = l: l].

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_ ? . . (5,5)
R730.

\ ..
v

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v 780l 4
v 6. (a) De?ne a convex set. Draw the region in the xy-plane where x + 2y S 6,
2x + y S 6 and x 2 0 y 2 0. Which corner minimizes the cost c = 2x ? y ?
v (b) Test whether the following matrix is positive de?nite or not.
7. (3) Find the best line b = C + Dt to ?t b = 0,8,8,20 at times t= 0,1,3,4.

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' ' . (b) Let V be the vector space generated by the three functions fl(x) = 1,
f2(x) = x, f3(x) = x2, where x e R. Let D : V ?) V be the derivative. What
is the matrix ofD w.r.t. the basis {fl, f2, f3} ? (5,5)
0?
(2000) ,.