
Linearity of Transformations; Basis and Dimensions for Four Subspaces
So its bases is {(1,4,5)^T} Linearity of Transformations and Basis and Dimensions for Four Subspaces are investigated in this solution.

Vectors and Matrices : Operations and Transformations
Find
a) u + v
b) u +  v 
c) Find two vectors in R³ with norm 1 orthogonal to be both u and v
d) Find norm of vector u /  u 
2) For which values of t are vectors u = (6, 5, t), and v = (1,t,t) orthogonal?

Linear Transformations, Rotations, Submodules and Subspaces
103616 Linear Transformations, Rotations, Submodules and Subspaces (7) Let F=R, let V=R^2 and let T be the linear transformation from V to V which is rotation clockwise about the origin by piradians.

Linear Transformations, Change of Basis and Conjugation
43661 Linear Transformations, Change of Basis and Conjugation Let V = Q3 and let ' be the linear transformation from V to itself:
'(x, y, z) = (9x + 4y + 5z,−4x − 3z,−6x − 4y − 2z), x, y, z E Q
With respect to the standard

Advanced Linear Algebra : Transformations, Basis and Eigenvalues
Transformations, Basis and Eigenvalues are investigated. The solution is detailed and well presented.

Simple Ring: Artenian Ring
First note that we can identify the elements of the ring A with the linear transformations VV. Moreover, such linear transformations are uniquely determined by their images on e_1, e_2,...,e_n.

Linear Algebra : Linear Transformations, Vector Space and Basis
Linear transformations, vector space and basis are investigated. The rank of matrices representatives are discussed.

The composition of linear maps and their underlying spaces.
Let U, V, and W be vector spaces over a field F. Suppose that T : U > V and S : V > W are linear transformations and that Im(T) = Ker(S). If T is injective and S is surjective, prove that
dim(V) = dim(U) + dim(W).

Kinetic energy, rest energy and inelastic collisions
Now pf dot pf is invariant under Lorentz transformations, so its value evaluated in its rest frame is the value in any frame. But in its own rest frame its velocity is zero. This means that P = 0 and E = M there and thus pf dot pf = M^2.