Category: MATHEMATICS

ARCHIVES 24 September 2014 1) Discuss for all values of k the system of equations 2x+3ky+(3k+4)z=0, x+(k+4)y+(4k+2)z=0, x+2(k+1)y+(3k+4)z=0.   2) Investigate for what values of α and µ the simultaneous equations x+y+z=6,x+2y+3z=10,x+2y+αz=µ have  (i) no solution ,(ii) a unique solution, (iii) an infinite number of solutions.     3) Show that the three equations -2x+y+z=a, …

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ARCHIVES 23 September 2014   1) Show that every square matrix is uniquely expressible as the sum of a symmetric and a skew-symmetric matrix.   2) If A is a square matrix of order n, prove that | Adj (Adj A) | =|A|(n-1)ᴧ2     3) Find the rank of the matrix 2  -2  0  …

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ARCHIVES 22 September 2014   MATRICES     1) Show that if a diagonal matrix is commutative with every matrix of the same order ,then it is necessarily a scalar matrix.   2) Find the possible square roots of the two rowed unit matrix I.   3) Show that the matrix B’AB is symmetric  or …

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ARCHIVES 20 September 2014 1) Find all (complex) characteristic values and characteristic vectors of the following matrices: 1 1 1 (b) 1 1 1 1 1 1             0 1 1 1 1 1             0 0 1         2) Let T  be the linear operator on R3 which is represented in the standard …

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ARCHIVES 19 September 2014   1) Let T be a linear operator on V3(R)  defined by T (a,b,c)=(3a,a-b, 2a+b+c)  for all (a,b,c)єV3(R).Is T invertible? If so, find a rule for T-1 like the one which defines T.   2) Let V(R) be the vector space of all polynomials in x with coefficients in R of …

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ARCHIVES 18 September 2014 1) Show that the mapping T:V2(R)―›V3(R) defined as  T(a,b)=(a+b,a-b,b) is a linear transformation from V2(R) into V3(R).Find the range,rank,null space and nullity of T.     2) Let T:R3―›R3 be the linear transformation defined by : T(x,y,z)=(x+2y-z, y+z, x+y-2z). Find a basis and the dimension of (i) the range of T; …

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ARCHIVES 17 September 2014   1) Show that the set S={1,x,x2,……,xn} of n+1 polynomials in x is a basis of the vector space Pn(R), of all polynomials in x (of degree at most n) over the field of real numbers.     2) “Corresponding to each subspace W1 of a finite dimensional vector space V(F),there …

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ARCHIVES 13 September 2014   Find whether  the vectors  2×3+x2+x+1, x3+3×2+x-2, x3+2×2-x+3 of R[x], the vector space of all polynomials over the real number field , are linearly independent or not.   Determine whether or not the following vectors form a basis of R3: (1,1,2),(1,2,5),(5,3,4).   Show that the vectors α1=(1,0,-1),α2=(1,2,1),α3=(0,3,-2) form a basis of …

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ARCHIVES 12 September 2014 1) Let V be the vector space of all functions from R into R; let  Ve be the subset of even functions , f(–x)=f(x); let Vo be the subset of odd functions, f(–x)=–f(x) Prove that Ve and Vo are subspaces of V. Prove that Ve+Vo=V. Prove that Ve ∩ V0= {0}. …

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