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Linear Algebra
Use Gram-Schmidt orthogonalisation process to find an orthonormal basis for the subspace of C^4 generated by the vectors (1,i,0,−i), (−i,0,1,2) and (0,−i,1,1).
Linear Algebra
Derive the matrix of translation in 2-dimensional plane? Why its first 2 column is identity matrix
Linear Algebra
Check whether the forms
2x^2+3y^2+5z^2−4xz−6yz
and 4x^2+3y^2+z^2−6xy−2xz
are orthogonally equivalent
2x^2+3y^2+5z^2−4xz−6yz
and 4x^2+3y^2+z^2−6xy−2xz
are orthogonally equivalent
Linear Algebra
Q. Derive the translation matrix? Why we take its first 3 column identify matrix.
Linear Algebra
A tank is fitted with two taps A and B of different size. If both the taps are opened simultaneously it takes 4 hours to fill the tank. If only tap A is opened for 1 hour and then only tap is opened for 4 hours the tank becomes half. Find the time taken by tap B alone to fill the tank.
A 6 hours
B 8 hours
C 12 hours
D 14 hours
A 6 hours
B 8 hours
C 12 hours
D 14 hours
Linear Algebra
Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) Subtraction is a binary operation on N.
ii) If{ v1, v2,..., vn} is a basis for vector space V,{ v1+ v2+···+ vn, v2, ..., vn} is also a basis for V.
iii) If W1 and W2 are subspaces of vector space V and W1+W2 =V, then W1∩W2 ={0}.
iv) The rank of a matrix equals its number of nonzero rows.
v) The row-reduced echelon form of an invertible matrix is the identity matrix.
vi) If the characteristic polynomial of a linear transformation is (x−1)(x−2), its minimal polynomial is x−1 or x−2.
vii) If zero is an eigenvalue of a linear transformation T, then T is not invertible.
viii) If a linear operator is diagonalisable, its minimal polynomial is the same as the characteristic polynomial.
ix) No skew-symmetric matrix is diagonalisable.
x) There is no matrix which is Hermitian as well as Unitary.
i) Subtraction is a binary operation on N.
ii) If{ v1, v2,..., vn} is a basis for vector space V,{ v1+ v2+···+ vn, v2, ..., vn} is also a basis for V.
iii) If W1 and W2 are subspaces of vector space V and W1+W2 =V, then W1∩W2 ={0}.
iv) The rank of a matrix equals its number of nonzero rows.
v) The row-reduced echelon form of an invertible matrix is the identity matrix.
vi) If the characteristic polynomial of a linear transformation is (x−1)(x−2), its minimal polynomial is x−1 or x−2.
vii) If zero is an eigenvalue of a linear transformation T, then T is not invertible.
viii) If a linear operator is diagonalisable, its minimal polynomial is the same as the characteristic polynomial.
ix) No skew-symmetric matrix is diagonalisable.
x) There is no matrix which is Hermitian as well as Unitary.
Linear Algebra
a) Which of the following functions are 1-1 and which are onto? Justify your answer.
i) f : R→R≥0 given by f(x) = x2 whereR≥0 is the set{x∈R|x≥0}.
ii) f : R→Rgiven by f(x) = x2+x+1.
b) If we consider the expression 1/(2−3x) as a function on R, what will be its domain and range? Will it have an inverse? Justify your answer.
c) Let a=( 1/2√2, √3/2√2, 1/√2)and b=(1/√2,0, 1/√2).
i) Find the direct cosines of aandb.
ii) Find the angle between aand b.
d) Check that the vectors u=(3 5, 4 5,0),v=(−4 5, 3 5,0) and w= (0,0,1) are orthonormal. Further, write the vector a= (1,−1,2) as a linear combination of the vectors.
i) f : R→R≥0 given by f(x) = x2 whereR≥0 is the set{x∈R|x≥0}.
ii) f : R→Rgiven by f(x) = x2+x+1.
b) If we consider the expression 1/(2−3x) as a function on R, what will be its domain and range? Will it have an inverse? Justify your answer.
c) Let a=( 1/2√2, √3/2√2, 1/√2)and b=(1/√2,0, 1/√2).
i) Find the direct cosines of aandb.
ii) Find the angle between aand b.
d) Check that the vectors u=(3 5, 4 5,0),v=(−4 5, 3 5,0) and w= (0,0,1) are orthonormal. Further, write the vector a= (1,−1,2) as a linear combination of the vectors.
Linear Algebra
a) Let V ={(a,b,c,d)∈R^4|a + b +2c + 2d = 0} and W ={(a,b,c,d)∈R^4|a= −b,c= −d}. Check that V and W are vector spaces. Further, check that W is a subspace of V.
b) Find the dimensions of V and W.
c) Let P^3 ={ax^3+bx^2+cx+d| a,b,c,d∈R}. Check whether f(x) = x^2+2x+1 is in [S], the subspace of P^3 generated by S ={3x^2+1, 2x^2+x+1}. If f(x) is in [S], write f as a linear combination of elements in S. If f(x) is not in [S], find another polynomial g(x) of degree at most two such that f(x) is in the span of S∪{g(x)}. Also write f as a linear combination of elements in S∪{g(x)}.
b) Find the dimensions of V and W.
c) Let P^3 ={ax^3+bx^2+cx+d| a,b,c,d∈R}. Check whether f(x) = x^2+2x+1 is in [S], the subspace of P^3 generated by S ={3x^2+1, 2x^2+x+1}. If f(x) is in [S], write f as a linear combination of elements in S. If f(x) is not in [S], find another polynomial g(x) of degree at most two such that f(x) is in the span of S∪{g(x)}. Also write f as a linear combination of elements in S∪{g(x)}.
Linear Algebra
Consider the following system of equations:
x1−3x2−x3 = 3
x1+5x2+3x3 = 1
−x1+7x2+3x3 = 1
Check whether the system of equations have a solution or not
x1−3x2−x3 = 3
x1+5x2+3x3 = 1
−x1+7x2+3x3 = 1
Check whether the system of equations have a solution or not
Linear Algebra
a) Let V be the vector space of polynomials with real coefficients and of degree at most 2.
If D = d/dx is the differential operator on V and B ={1+2x^2,x+x^2,x^2} is an ordered basis of V,
find [D]B. Find the rank and nullity of D. Is D invertible? Justify your answer.
b) Let T: R^2 →R^2 and S: R^2 →R^2 be linear operators defined by T (x1,x2) = (x1+x2, x1−x2) and S(x1, x2) = (x1, x1+2x2) respectively.
i) Find T◦S and S◦T.
ii) Let B ={(1,0),(0,1)}be the standard basis of R3. Verify that [T◦S]B = [T]B◦[S]B.
c) Find the inverse of the matrix 1 −1 0
2 −1 1
1 1 −1 using row reduction.
d) Let B1 ={(1,1),(1,2)}and B2 ={(1,0),(2,1)}. Find the matrix of the change of basis from B1 to B2.
If D = d/dx is the differential operator on V and B ={1+2x^2,x+x^2,x^2} is an ordered basis of V,
find [D]B. Find the rank and nullity of D. Is D invertible? Justify your answer.
b) Let T: R^2 →R^2 and S: R^2 →R^2 be linear operators defined by T (x1,x2) = (x1+x2, x1−x2) and S(x1, x2) = (x1, x1+2x2) respectively.
i) Find T◦S and S◦T.
ii) Let B ={(1,0),(0,1)}be the standard basis of R3. Verify that [T◦S]B = [T]B◦[S]B.
c) Find the inverse of the matrix 1 −1 0
2 −1 1
1 1 −1 using row reduction.
d) Let B1 ={(1,1),(1,2)}and B2 ={(1,0),(2,1)}. Find the matrix of the change of basis from B1 to B2.
