Linear Algebra Answers

Apply the Gram-Schmidt orthogonalisation process to find an orthonormal basis for the subspace of R4 generated by the vectors
{(-1,1,0,1),(1,0 ,-1,0),(1,0,2,-1)}

Let T: R4-R4 be defined by
T(x1,x2,x3,x4)=(-x2,x1,-x4,x3)
Check whether T is a linear operator and T4=I. Is T invertible?

Consider the linear operator T : C
4 → C
4
, defined by
T (z1,z2,z3,z4) = (−iz2,iz1,−iz4,z3).
i) Compute T
∗
and check whether T is self-adjoint.
ii) Check whether T is unitary.

Reduce the conic x
2 −6xy+y
2 −4 = 0 to standard form. Hence the given conic.

Which of the following sets are convex? Give reason.
(i) A={(x1, x2):x1,x2 ≤1; x1, x2 ≥0}
(ii) B={(x1, x2):x2-3≥x1²; x1, x2 ≥0}

An electronics company produces​ transistors, resistors, and computer chips. Each transistor requires 3 units of​ copper, 1 unit of​ zinc, and 2 units of glass. Each resistor requires 3​, 2​, and 1 units of the three​ materials, and each computer chip requires 2​, 1​, and 2 units of these​ materials, respectively. How many of each product can be made with 1570 units of​ copper, 740 units of​ zinc, and 880 units of​ glass? Solve this exercise by using the inverse of the coefficient matrix to solve a system of equations.

Let V be the vector space of 2× 2 matrices over R. Check whether the subsets
W1 = { (a,1) ,(0,-a) | a ∈ R} and W2 = { (a,-a), (0,b)| a,b∈ R}
are subspaces over R. For those sets which are subspaces, find their dimension and a
basis over R.

Let V be a vector space over a field F and let T : V → V be a linear operator. Show
T(W) ⊂ W for any subspace W of V if and only if there is a λ ∈ F such that Tv = λv
for all v ∈ V .

1) Which of the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.
i) The relation ∼ defined by R by x ∼ y if x ≥ y is an equivalence relation.
ii) If S1 and S2 are finite non-empty subsets of a vector space V such that [S1] = [S2], then
S1 and S2 have the same number of elements.
iii) For any square matrix A, ρ(A) = det(A)
iv) The determinant of any unitary matrix is 1.
v) If the characteristic polynomials of two matrices are equal, their minimal polynomials are
also equal.
vi) If the determinant of a matrix is 0, the matrix is not diagonalisable.
vii) Any set of mutually orthogonal vectors is linearly independent.
viii) Any two real quadratic forms of the same rank are equivalent over R.
ix) There is no system of linear equations over R that has exactly two solutions.
x) If a square matrix A satisfies the equation A2 = A, then 0 and 1 are the eigenvalues of
A.

Which of the following are binary operations on S = {x ∈ R | x > 0}? Justify your
answer.
i) The operation Δ defined by xΔy = x(y− 2).
ii) The operation ∇ defined by x∇y = e^x+y.
Also, for those operations which are binary operations, check whether they are
associative and commutative.