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

Sr. No. of Question Paper : 7801 F-2 Your Roll No................

Unique Paper Code : 2341201

Name of the Course : B.Tech. in Computer Science

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Name of the Paper : Linear Algebra for Comp. Sc.

Semester : II

Duration: 3 Hours Maximum Marks: 75

Instructions for Candidates

1. Write your Roll No. on the top immediately on receipt of this question paper.

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2. Attempt five questions in all.
3. Question No. 1 is compulsory.
4. All the symbols have their usual meanings.

1. (a) Find the inverse of the following matrix using elementary row transformations
   $$\begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & -1 \\ 3 & -5 & 0 \end{bmatrix}$$
   

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   (b) Describe the null space of the matrix A = $$\begin{bmatrix} 1 & 2 & 2 & 4 \\ 3 & 8 & 6 & 16 \end{bmatrix}$$
   (c) State and prove Cauchy-Schwarz's inequality.
   (d) Use Cramer's rule to solve Ax = b, where
   $$A = \begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{bmatrix} \text{ and } b = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$$
   (e) Find the Eigen values and the corresponding Eigen vectors of the matrix
   

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   $$A = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
   (f) Let M be the space of all n x n matrices. Let T: M? M be a map such that
   $$T(A) = \frac{A - A^t}{2}$$
   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 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. (5×7=35)

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2. (a) Find the complete solution to Ax = b, where
   $$A = \begin{bmatrix} 1 & 3 & 0 & 2 \\ 0 & 0 & 1 & 4 \\ 1 & 3 & 1 & 6 \end{bmatrix} \text{ and } b = \begin{bmatrix} 1 \\ 6 \\ 7 \end{bmatrix}$$
   (b) Write the LU-decomposition of the matrix
   $$A = \begin{bmatrix} 3 & -6 & -3 \\ 2 & 0 & 6 \\ -4 & 7 & 5 \end{bmatrix}$$
   Also compute E₁₁, E₂₁, and E₃₂ to find L. (5,5)

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3. (a) Define 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^m and Rⁿ respectively.
   (b) Find the bases and dimensions for all four fundamental subspaces of the following matrix A, namely, the column space of A, the null space of A, the row space of A and the null space of A^T. Also determine the rank of the matrix A.
   $$A = \begin{bmatrix} 1 & 3 & 1 & 2 \\ 2 & 6 & 4 & 8 \\ 0 & 0 & 2 & 4 \end{bmatrix}$$ (5,5)
4. (a) Use Gram-Schmidt's orthogonalisation process to find the QR decomposition of the matrix
   $$W = \begin{bmatrix} 1 & 1 & 1 \\ 1 & -2 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 2 \end{bmatrix}$$
   

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   (b) Find the pivots of the given matrix A and verify that det. A = product of the pivots.
   $$A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 2 & 3 \\ 3 & 3 & 3 \end{bmatrix}$$ (5,5)
5. (a) Diagonalise the following matrix A and hence find A³.
   $$A = \begin{bmatrix} 3 & 4 \\ 4 & -3 \end{bmatrix}$$
   (b) Find the singular value decomposition (svd) of the matrix FirstRanker.com
   

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   $$A = \begin{bmatrix} 2 & 2 \\ 1 & 1 \end{bmatrix}$$ (5,5)
6. (a) Define a convex set. Draw the region in the xy-plane where x + 2y = 6, 2x + y = 6 and x = 0 y = 0. Which corner minimizes the cost c = 2x - y ?
   (b) Test whether the following matrix is positive definite or not.
   $$A = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}$$ (5,5)
7. (a) Find the best line b = C + Dt to fit 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 f₁(x) = 1, f₂(x) = x, f₃(x) = x², where x ? R. Let D : V ? V be the derivative. What is the matrix of D w.r.t. the basis {f₁, f₂, f₃} ? (5,5)

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