Skip to main content

MTE-02 Linear Algebra Solved Assignment For IGNOU Maths 2018 Session

IGNOU MTE-02 Solved Assignment For 2018 Session

IGNOU Maths Solved Assignment for 2018

MTE-02 Linear Algebra

      1. Which of the following are binary operations on S = { x ∈ R | x > 0 } ? Justify your answer.
      2. The operation 4 defined by x 4 y = x 2 + y 3 . Also, for those operations which are binary operations, check whether they are associative and commutative.
    1. Check whether the vector ( 2 √ 3 , 2 ) is equally inclined to the vectors ( 2 , 2 √ 3 ) and ( 4 , 0 )
    2. Find the radius and the center of the circular section of the sphere | r | = 26 cut off by the plane r · ( 2 i + 6 j + 3 k ) = 70
    1. This Question(2a) is not solved yet
    2. Complete the set S = { x 3 + x 2 + 1 , x 2 + x + 1 , x + 1 } to get a basis of P 3
    3. Check whether each of the following subsets of R 3 is linearly independent. i) { ( 1 , 2 , 3 ) , ( − 1 , 1 , 2 ) , ( 2 , 1 , 1 ) } . ii) { ( 3 , 1 , 2 ) , ( − 1 , − 1 , − 3 ) , ( − 4 , − 3 , 0 ) } .
    4. Let V = R 3 and W = { ( x , y , z ) | x + y + z = 0 } be a subspace of V . Which of the following pairs of vectors are in the same coset of W in V ? i) ( 1 , 3 , 2 ) and ( 2 , 2 , 2 ) . ii) ( 1 , 1 , 1 ) and ( 3 , 3 , 3 ) .
    1. Let V = R 3 , A = { ( x , y , z ) | y = 0 } and B = { ( x , y , z ) | x = y = z } . Check whether R 3 = A ⊕ B
    2. Let T : R 3 → R 3 be defined by T ( x 1 , x 2 , x 3 ) = ( x 1 − x 3 , x 2 − x 3 , x 1 ) . Is T invertible? If yes, find a rule for T − 1 like the one which defines T .
    3. Find the inverse of the matrix   − 1 2 1 0 1 1 1 0 2   using row reduction.
    1. Check whether the following system of equations has a solution.
      4 x + 2 y + 8 z + 6 w = 3
      2 x + 2 y + 2 z + 2 w = 1
      x + 3 z + 2 w = 3
    2. Define T : R 3 → R 3 by T ( x , y , x ) = ( x + y , y , 2 x − 2 y + 2 z ) . Check that T satisfies the polynomial ( x − 1 ) 2 ( x − 2 ) . Find the minimal polynomial of T .
    1. Let T : P 2 → P 1 be defined by T ( a + bx + cx 2 ) = b + 2 c +( a − b ) x .
      Check that T is a linear transformation. Find the matrix of the transformation with respect to the ordered bases B 1 = { x 2 , x 2 + x , x 2 + x + 1 } and B 2 = { 1 , x } . Find the kernel of T .
    2. Consider the basis e 1 = ( − 2 , 4 , − 1 ) , e 2 = ( − 1 , 3 , − 1 ) and e 3 = ( 1 , − 2 , 1 ) of R 3 over R . Find the dual basis of { e 1 , e 2 , e 3 } .
    1. Check whether the matrices A and B are diagonalisable. Diagonalise those matrices which are diagonalisable.
    2. Find inverse of the matrix B in part a) of the question by finding the adjoint as well as using Cayley-Hamiltion theorem.
    1. Let P 3 be the inner product space of polynomials of degree at most 3 over R with respect to the inner product 〈 f , g 〉 = ∫ 1 0 f ( x ) g ( x ) dx . Apply the Gram-Schmidt orthogonalisation process to find an orthonormal basis for the subspace of P 3 generated by the vectors { 1 − 2 x , 2 x + 6 x 2 , − 3 x 2 + 4 x 3 }
    2. Consider the linear operator T : C 3 → C 3 , defined by T ( z 1 , z 2 , z 3 ) = ( z 1 − iz 2 , iz 1 + 2 z 2 + iz 3 , − iz 2 + z 3 ) .
      i) Compute T ∗ and check whether T is self-adjoint.
      ii) Check whether T is unitary.
    3. Let ( x 1 , x 2 , x 3 ) and ( y 1 , y 2 , y 3 ) represent the coordinates with respect to the bases B 1 = { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) } , B 2 = { ( 1 , 0 , 0 ) , ( 0 , 1 , 2 ) , ( 0 , 2 , 1 ) } . If Q ( X ) = x 2 1 + 2 x 1 x 2 + 2 x 2 x 3 + x 2 2 + x 2 3 , find the representation of Q in terms of ( y 1 , y 2 , y 3 )
    1. Find the orthogonal canonical reduction of the quadratic form x 2 + y 2 + z 2 − 2 xy − 2 xz − 2 yz . Also, find its principal axes.
    2. Reduce the conic x 2 + 6 xy + y 2 − 8 = 0 to standard form. Hence identify the given conic.
  1. Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
    1. R 2 has infinitely many non-zero, proper vector subspaces.
    2. If T : V → W is a one-one linear transformation between two finite dimensional vector spaces V and W then T is invertible.
    3. If A k = 0 for a square matrix A , then all the eigenvalues of A are zero.
    4. Every unitary operator is invertible.
    5. Every system of homogeneous linear equations has a non-zero solution.

IGNOU BDP Solved Assignmets For All Semesters

Comments

Popular This Week

IGNOU BA (BDP) Solved Assignments For All Subjects 2017-18 Session

MCO-05 Solved Assignment For IGNOU MCOM 2nd Year 2018 Session

IGNOU MCOM Solved Assignment for 2018 (FREE) MCO-05 Accounting For Managerial DecisionsQuestions Solved In Assignment:“Balance Sheet is a statement of assets and liabilities or sources and uses of funds or both”, Comment.“Funds flow statement is only supplementary to P/L Account and Balance Sheet, it can not substitute to P/L Account and Balance Sheet”. Do you agree to this statement? Explain your views.Comment on the following statement: A budget is both a plan as well as a control tool.Budget should be regarded as a master but not as a servant.Fixed budget is more useful than a flexible budget.Performance budgeting lays immediate stress in the achievement of specific goals over a period of time.Describe various sectors of chemical industry. What are India’s competitive advantages & disadvantages in the export of chemical goods?Differentiate between the following: Standard cost and Estimated costFixed overhead Efficiency variance and fixed overhead calendar varianceMarginal costi…

BEGE-101/EEG-01 Solved Assignment For IGNOU BDP 2017-18

IGNOU BDP Solved Assignment for 2018 (FREE)BEGE-101/EEG-01 Elective Course In English SOLVED ASSIGNMENT Read the following poem carefully and answer the following questions. Amalkanti (Nirendranath Chakrabarti) Amalkanti is a friend of mine, we were together at school. He often came late to class and never knew his lessons. When asked to conjugate a verb, he looked out of the window in such puzzlement that we all felt sorry for him. Some of us wanted to be teachers, some doctors, some lawyers. Amalkanti didn't want to be any of these. He wanted to be sunlight — the timid sunlight of late afternoon, when it stops raining and the crows call again, the sunlight that clings like a s mile to the leaves of the jaam and the jaamrul. Some of us have become teachers, some doctors, some lawyers. Amalkanti couldn't become sunlight. He works in a poorly lit room for a printer. He drops in now and then to see me, chats about this and th at over a cup of tea, then gets up to go. I see him of…

BSHF-101 (ENGLISH MEDIUM) SOLVED ASSIGNMENT FOR IGNOU BDP

Download Assignment IGNOU BDP Solved Assignment for 2018 (FREE)BSHF-101 Foundation Course in Humanities and Social SciencesQUESTIONS SOLVED IN ASSIGNMENT:Discuss the different aspects of the non-cooperation movement. Discuss some of the sources of thinking that went in to formulating a strategy for development immediately after independence? What do you understand by the term 'renaissance'? How did the Indian economy adapt to globalisation? Comment. Comment on the main environmental challenges we are facing today?What do you understand by the term 'Fundamental Duties' as given in our constitution? SCQ: Write short notes on any two in about 100 word s each: BharatnatyamSwara in Indian musicDownload Assignment

MTE-01 Calculus Solved Assignment For IGNOU Maths 2018 Session