Linear Algebra Answers

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

(a) Let A be a square matrix and fA(x) (x = lambda) its characteristic polynomial. In each of the following cases (i) to (iv), write down whether
(D) A is diagonalizable over R
(N) A is not diagonalizable over R
(U) it is not possible to say one way or the other.
(i) fA(x) = (x - 3)^2(x - 5)
(ii) fA(x) = (x^2 - 1)(x^2 - 2)
(iii) fA(x) = (x^2 + 6)(x - 1)(X + 2)
(iv) fA(x) = (x^2 + 3)^2
(b) For each case that you marked (N), say whether or not the matrix can
certainly be diagonalized over C

Let A = (-2 -1 -3 6 1 // 1 1 1 -2 1 // 2 3 1 -2 4 ) Find a basis for the column space Col(A) of A, and a basis for the null spaceNul(A) of A.Show that for any matrix A, if u and v are in the null space of A, then
so is u + v.

[b] u[/b] and [b]v[/b] are two non null vectors in [b]R[sub]n[/sub][/b],&
||u|| and ||v|| denote their respective lengths, and.&
||u – v|| denotes difference between u and v.
If ||u||=1 and ||v||=1 and ||u – v|| = ,& & then the u and v are&
& i. orthogonal&
& ii. Linearly independent&
& iii. Both i. and ii.&
& iv. All of them