Linear Algebra Answers

Suppose x, y are elements of R^n and are nonparallel vectors.

a) Prove that if sx + ty = 0, then s = t = 0.
b) Prove that if ax + by = cx + dy, then a = c and b = d.

Let a,b and c be three vectors such that a + b + c = 0. Given that a x b = b x c = c x a. Explain what this means geometrically. (Your answer should refer to a triangle)

Let A=1/3 (-2 -1 2
2 -2 1
1 2 2)
Prove that A is the product of a rotation and a reflection. Prove that A is an orthogonal matrix.

B=(x[sup]2[/sup]+x, x[sup]2[/sup]-2, x[sup]2[/sup]+2x-1) is a subset of the vector space P2 of polynomials of degree no larger than two,T(x[sup]2[/sup]+x) =(1,-2) T(x[sup]2[/sup]-2) = (4,1)& & T(x[sup]2[/sup]+2x-1) = (2,-1)what is matrix representation for T with respect to the bases B for P2 and S=(1,0)(0,1) for R[sup]2[/sup]?
With T given as in the above question, calculate T(7x[sup]2[/sup]+3x-2).

Consider the set of all functions [0,1] which are discontinuous. check if this set is a vector space over R with respect to pointwise addition and scalar multiplication

Find the direction cosines of the perpendicular from the origin to the plane 3r.(2i − 3j+ k) + 7 = 0.

If u = 2i − 3j + k, v = i − 2k, w = ai + bj + ck form an orthogonal basis of R3 , find the
possible values of a, b and c.
Further, obtain the angles x = i − 3j− 3k makes with each of these vectors.
b) Find the direction cosines of the perpendicular from the origin to the plane
3r.(2i − 3j+ k) + 7 = 0.

1/ Prove that the set V=R+ ( the set of all positive real numbers) is a vector space with the following nonstandard operations: for any x,y belong to R+ & for any scalar c belong to R:
x O+ ( +signal into circle) y=x.y (definition of vector addition) & c O ( dot signal into circle) x = x^c (definition of scalar multiplcation) ( must verify that all 10 axioms defining a vectorspace are satisfied).
2/ Consider the vector space V = C (-infinite,infinite)= all fumctions f(x) which are continuous everywhere. Show that the following subset H of C (-infinite, infinite) is in fact a subspace of C (-infinite,infinite):
H= {all functions f(x) satisfying the differential equation f " (x) +25 f(x)=0}
(need to verify all 3 subspace requirements)

A = 4 2
-5 -2

Find an invertible 2x2 matrix Q such that Q^-1AQ is equal to a rotation followed by scaling by a positive real number, and write down both the rotation matrix and the scaling matrix. What is the angle of rotation of the matrix?

Find the (complex) roots of f(x) by solving f(x) = 0 directly, and deduce that w = (-1 + i (sqrt3))/2