Linear Algebra Answers

1) given that A1=2i-j+k, A2=i+3j-2k, A3=3i+2j+5k, and A4=3i+2j+5k
find scalars a,b,c such that A4=aA1+bA2+cA3.

2) if a and b are non-collinear vectors and A=(x+y)a+(2x+y+1)b

3) given the scalar defined by phy(x,y,z)=3x^2-xy^2+5

Let V be the set of all functions that are twice differentiable in R and S={cosx,sinx,xcosx,xsinx}. a)Check that S is a linearly independent set over R.(Hint: Consider the equation a0cosx+a1sinx+a2xcosx+a3xsinx. Putx=0,π,π 2 ,π 4 ,etc.and solve for ai.) b) Let W=[S]and let T:V→V be the function defined by T(f(x))=d2 dx2(f(x))+2d dx(f(x)). Check that T is a linear transformation on V.

Is there a solution for AX=B matrix equation with zero diagonal constraint, such that: X_ii=0 ?
where, A, B, X are n×n matrices.

Is it right to solve the equation as follows?

X=inv(A)*B
X_ii=0

Under what condition A.B is not equal to zero and A×B is equal to zero when A and B are two non zero vectors?

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.

question by finding the adjoint as well as
using Cayley-Hamiltion theorem

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}