BSc Mathematics SEM II ATKT 2018-19 ATKT Mathematics II Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate full marks

1. Q1 Choose the correct alternative in each of the following: 20 marks
   • i. For which value of k does the following system have infinitely many
   • ii. Which of the following matrix is skew symmetric?
   • iii. The parametric representation for the line through the points (1,2) and
   • (c) (d) None of these
   • iv. Which of the following sets is linear independent? Which of the following is not a subspace of R* over R?
   • vi. of the following is a generating set of R*? The rank of linear transformation > R? defined as
   • viii. The nullity of the linear transformation > defined as
   • ix. If for a linear transformation T: > R* the Dim(Ker(T)) = 1 then the Rank of T is
   • x. of the following is the basis for
2. Q2 Attempt any ONE question from the following: 8 marks
   • i. If A and B are n X n matrices then prove that
   • (2) If Ais invertible prove that A’ is invertible and
   • ii. using induction on m prove that any homogeneous system of linear equations in n unknowns has a non trivial solution if m <n
   • b) Attempt any TWO questions from the following: Let be n n real matrix and X = (x1, If a= y an) and B = Bn) are solutions of the linear homogeneous system AX = 0, then prove that a+ 6 and ka, a R are also solutions of the 12
   • ii. Use parametric equations of line to check if the points (1, (2, 0, —3)and (4, 4, - 6) are collinear
   • iii. Geometrically interpret solutions of the real linear homogenous system of 2 equations in 3 unknowns
   • iv. Solve the system of linear equations: x + y+z=3,
3. Q3 a) Attempt any ONE question from the following: 8 marks
   • i. Prove the following properties of a real vector space V: 2.0.v=0,, Vv EV li. vector space and = that S is linear dependent if and only if one of the vectors in S can be written linear combination of the other vectors in S
   • b) Attempt any TWO questions from the following: that W is vector subspace of V 12
   • ii. Check whether {(1,3), (4, 0), (9,15)} is a linear independent set Prove. the following properties of a real vector space:
4. Q2 Every vector in V has a unique additive inverse
   • iv. linear span of a non-empty subset of a real vector Express vector x? in V asa linear combination of Attempt any ONE question from the following: (08)
   • i. Show that every finitely generated vector space has a basis
   • ii. Let V and W be real vector spaces over and T:V > bea linear transformation. Prove that KerT is a subspace of V and Image T is a subspace of W
   • b) Attempt any TWO questions from the following: 12
   • i. Check if the set { (1,0,1), (1,1,0), is a basis of R?
   • ii. Find the dimension of image space of T: > defined by
   • iii. | Find the matrix associated with the linear transformation T: R? > R* defined by T(x, y,z) = (x + y, x,y,z) with respect to standard bases of R? and
   • iv. Find the basis of the subspace W = R? | x+y = 0} and extend it to.a basis of
5. Q5 Attempt any FOUR questions from the following: 20 marks
   • a) Find the value of so that the system of linear equations
   • b) Transform the following matrix to it’s row echelon form :
   • c) LetV space of all real valued sequences and S= {(x,) ts convergent }. Show that S is a subspace of
   • d) whether the set S generates where Alinear transformation T : R? — R? is such that
   • f) the Rank-Nullity theorem for T : R? > R defined as

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