Let v and w be vector spaces and t: v –> w be linear.

Let V and W be vector spaces and T: V --> W be rectilinear. Deem that T is one-to-one and that S is a subset of V. Prove that S is rectilinearly defiant if and solely if T(S) is rectilinearly defiant.

Relevant Theorems:

(1) T: V --> W is rectirectistraight if T(x+y) = T(x) + T(y) and T(cx) = cT(x)

(2) nullspace (or meat) N(T) = {x in V : T(x) = 0}

(3) order (or fiction) R(T) = {T(x) : x in V}

(4) Let V and W be vector spaces and T: V --> W be rectilinear. Then N(T) and R(T) are subspaces of V and W respectively

(5) Let V and W be vector spaces and T: V --> W be rectilinear. If B = {v1, v2, v3, .... , vn} is a plea for V, then

R(T) = brace{ T(B) = brace{ T(v1), T(v2), T(v3), ..... , T(vn) } }

(6) If N(T) and R(T) are terminable-dimensional, then nullity(T) = dim[ N(T) ], and systematize(T) = dim[ R(T) ]

(7) Mass Theorem: Let V and W be vector spaces and T: V --> W be rectilinear. If V is terminable-dimensional, then

nullity(T) + systematize(T) = dim(V)

(8) Let V and W be vector spaces, and let T: V --> W be rectilinear. Then T is one-to-one if and solely if N(T) = {0}, and T is onto if and solely if dim{R(T)} = dim(W)

(9) Let V and W be vector spaces of correspondent (finite) mass, and T: V --> W be rectilinear. Then the subjoined are equivalent:

(i) T is one-to-one

(ii) T is onto

(iii) systematize(T) = dim(V)

(10) Let V and W be vector spaces aggravate F, and deem that {v1, v2, ... , vn} is a plea for V. For w1, w2, ... , wn in W, there exists accurately one rectirectistraight transmutation T: V --> W such that T(vi) = wi for i = 1, 2, ... , n.

(11) Let V and W be vector spaces, and deem that V has a terminable plea {v1, v2, ... , vn}. If U, T: V --> W are rectirectistraight and U(vi) = T(vi) for i = 1, 2, ... , n, then U = T

Note: I CANNOT use the subjoined (a former conclusion): "T is one-to-one if and solely if T carries rectilinearly defiant subsets of V onto rectilinearly defiant subsets of W".