Linear Algebra/Linear Algebra (LA) BS 4 Semester/Term University Of Sargodha (UOS) 2023

University of Sargodha

BS 4'' Term Examination 2023

Subject: Computer Sciences

Time Alloyed: 02:30 hours

Paper: Linear Algebra (MATH-202/MATH-3215)

Maximum Marks: 60

Q.1.

Q.2.

Q.3.

Q.5.

Objective Part

(Compulsory)

Write short answers of the following in 2-3 lines each on your answer sheet.

(2*12)

(u,0)

1. If V is an inner product space and u, e eV. Then show that cost =

4i.

iii.

Define similarity of matrices.

Define Hermitian matrix with an example.

Convert the coefficient matrix of the given system into echelon form;

x-*2+2x=0.4x+x2+2xg = 1,x+x2+ x3 = -1.

Find the eigenvalues of coso

-sin0

sine

cose

vi.

Check whether W = ((x,y, z) e R$. 2x + 3y - 4z =

0) a subspace of R3.

Find the dimension of the subspace (x 2, 23, x4): x2 = xg) of R*

Define diagonalization of a matrix.

Determine whether or not the set of vectors ((1,1), (3,1)) is basis of R?

Let V be the real space of all functions defined on R. Check x, cosx in V are linearly dependent

or linearly independent.

xi.

Find an orthogonal matrix, whose first row is %)

and uz =

xii.

Find a vector orthogonal to 4 =

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Subjective PaR

(3*12)

IA = |1

3]. Find a matriz such that AB =

12. Does this mean A is inverible? Explain.

if A and B are 6 x 6 matrices such that det(182) = 72 and (1283) attend re(24) and.

Define subspace of a vector space. Find an equation of the subspace W of R? spanned by set of

vectors ((1, -3,2). (-2,0,3)).

Suppose that U and W are distinct four dimensional subspaces of a vector space Vof dimension

six. Find the possible dimension of U n W.

Show that {(1,0,1), (0,1,1), (0,0,1)) is a basis of R'. Find an orthonormal basis of R° using the

Gram-Schmidt process.

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