Mind-Warm-Ups Vector Spaces

Ultimate Space = S
si = arbitrary partition of S
Δsi = infinitesimal si, that is limit of si as it tends to zero.
pi = arbitrary point in S = Δsi

Problem 21: let X and Y be linear spaces (vector spaces) of the same field F, and α1, α2 scalars in F. What criteria determine whether or not a transformation T from X into Y is a linear transformation (also called a linear operator)?
Ans: if T is a linear transformation, the following condition must hold:
T(α1x1 + α2x2) = α1T(x1) + α2T(x2)

Problem 22: let X and Y be linear spaces over the same field F. The dimension of X is n; the dimension of Y is m. If m = n, then X and Y are isomorphic. True or False?
Ans: True. If the linear transformation T(x) = 0 implies x = 0, then T is a one-to-one linear transformation. If T is a one-to-one linear transformation from the linear space X into the linear space Y, then X and Y are isomorphic and have equal dimensions.

Problem 23: identify the following space:
W = {x ∈ X:T(x) = 0}; where T is a linear transformation from the linear space X into the linear space Y.
Ans: W is a subspace of X and it is called the null space or kernel of T.

Problem 24 : is the transformation T(x,y) = (x, -y) from V2(R) into V2(R) a linear transformation?
Ans: let u1 = (x1, y1);
and u2 = (x2, y2) in the domain of T. Let a , b be scalars in the real field R.
au1 = a(x1, y1); bu2 = b(x2, y2)
T(au1 + bu2) = (ax1 + bx2 - ay1 - by2)
aT(u1) + bT(u2) = (ax1 + bx2 - ay1 - by2)
Therefore; T is a linear transformation. This is a straight forward approach. A conceptual approach is as follows:
Let u = (x,y); T(x,y) = T(u) = Au
T(au1 + bu2) = A(au1 + bu2)
=aAu1 + bAu2 = aT(u1) + bT(u2)
Therefore, T is a linear transformation. T(u) is the reflection of u in the x axis. In linear algebra, A is a matrice. We shall see it again in the concept of Identity.

Problem 25 : what is an Identity transformation?
Ans: if T(u) = u is a linear transformation and its domain and its range are equal, then T is called an identity transformation.

Problem 26: let T be a linear transformation from Vn(R) into Vn(R) such that v = T(u), where u is a vector from the domain of T and v is a vector from the range of T.
(i) what is the inverse transformation T-1 of T?
(ii) Is T-1 a linear transformation?
Ans: (i) u = T-1(v). The domain of T, u is the range of T-1 and the range of T is the domain of T-1.
(ii) try it.

Problem 27: if T is a linear transformation from R1 to R1 what is a general expression for T?
Ans: T(x) = cx.

Problem 28: is the transformation T(x,y) = (x, 3y) from R2 to R2 a linear transformation?
Ans: yes. If the vector u =(x,y), there is an A for which T(u) =Au. Therefore T is a linear transformation. The vertical points are expanded by 3. This problem is an example of a 2-dimensional scaling transformation.

Problem 29: the dot or scalar product of the pair of vectors u = u1,...un; and v = v1,...vn; in Vn(R) is defined as u.v = u1v1 + u2v2...+ unvn = |u||v|cosθ, where θ is the angle between them. Show that the standard basis i = (1,0,0), j = (0,1,0) and k = (0,0,1) of V3(R) are mutually perpendicular.
Ans: try it. Hint: determine that the dot product for each pair of basis = 0, then equate |u||v|cosθ to 0.

Problem 30: if T is the transformation T(x,y) = (xcosθ + ysinθ, xsinθ + ycosθ) from R2 to R2 (θ in radians). What type of transformation is T and is T a linear transformation?
Ans: T is a rotation of a vector u in R2, θ degrees counterclockwise about the y axis. Yes T is a linear transformation. Solution will be clear after familiarization with matrices.

Peter Oye Sagay