BSc CS Sem 3 BSc CS Semester 3 (2022 2023) Oct 2023 LINEAR ALGEBRA Question Paper - Mumbai University | munotes

Questions asked in this paper

• (ii) Figures to the right indicate marks

1. Q1 Attempt the following (Any Four) (20M)
   • a) Define Modulus and argument of complex number z. calculate modulus and argument
   • b) Prove all five addition properties of vector space for the set S = {(x, R}
   • c) Explain the rotation of complex number. Also if z = 2+4i, then find coordinate of z in
   • d) Solve the non-homogenous linear equations
   • e) Find the square root of z = 21 — 20i
   • f) Define linear combination of vectors in vector space V. Verify (6,5,3) as a linear combination of (1,0,0), (1, 1,0) and (1,1,1)
2. Q2 Attempt the following (Any Four) (20M)
   • a) Let T: IR? > R defined by T(x, y) = x — y. Show that T is a linear Transformation
   • b) Find dimension of the space spanned by vectors (1,0,3), (0,4,0), (0,0,3), (2,1,3) and also find the basis
   • c) Using Gauss elimination method solve the following cquation
   • d) Define linearly dependent and independent vectors. Verify S = {(1,0,0),(1,4,0), (1,4,6)} is linearly dependent or independent
   • e) Reduce the following matrix in a Row-Echelon form
   • f) Define subspace of a vector space V Verify the set W= {( x,0)/x JR} is subspace of /R?
3. Q3 Attempt the following (Any Four) (20M)
   • a) Find Eigen values and Eigen vectors for the matrix alo 3
   • b) Use the Gram-Schmidt orthonormalization process to construct an orthonormal set of vectors ‘rom the linearly independent set {(1,0,2),(-1,0,1)}
   • c) Let X = and Y= be two vectors in /R* then prove < X,Y + is an inner product in /R? VCD/ SYCS Algebra 75 MARKS
   • d) Find minimal polynomial m(t) of 7 —15
   • e) Check whether the set G , 0), (0,0,1)} is orthonormal
   • f) Verify Cayley Hamilton Theorem for matrix
4. Q4 Attempt the following (Any Five) (15M)
   • a) Check whether the set {(1,2,1),(4,-2,0),(2,4,-10)} is orthogonal
   • b) Consider the basis B = and B’ = {(1,0), (0,1)}. If u is a vector such that ug = [3 then find ug
   • c) Verify the set a,b,c IR} is a vector space in IR with respect addition and Scalar multiplication
   • d) Find distance and angle between two vectors x= (1,2,3) and y=(3,-2,1) in
   • e) Express the number in the polar form
   • f) Find vector-matrix multiplication in terms of linear combination for

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