Linear Algebra Answers

Find the radius and the center of the circular section of the sphere |r| = 17 cut off by
the plane r . (i + 2j + 2k) = 24.

Find the equation of the line passing through the point A(1, 0, -1) and parallel to the
line joining B(1, 2, 3) and C( -1, 2, 0). Is it perpendicular to the line
(1 + 3α , 0 ,1 - 2α ).

Let V=R2 . Define addition + on V by ( x1 , y1 ) + ( x2 , y2 ) = ( x1+x2 , y1+y2 ) and scalar multiplication . by r . (a , b) = (ra , 0). Check whether V satisfies all the
conditions for it to be a vector space over R with respect to these operations.

use the Gaussian Elimination method to find the value of a so that the system of equations.
x+(a+4)y+(4a+2) z=0
2x+3ay+(3a+4) z=0
x+2(a+1)y+(3a+4) z=0
has (i) a unique solution ;
(ii) infinitely many solutions.
Further, find the solution set in each case.

If A B C are three vectors show that A*(B+C)=A*B+B*C

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

a) x=2,y=4
b) x=2,y=1
c)x=4,y2
d)x=1,y=1

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?