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## 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.

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