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Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

Ultimate Space = S

s_{i} = arbitrary partition of S

Δs_{i} = infinitesimal s_{i}, that is limit of s_{i} as it tends to zero.

p_{i} = arbitrary point in S = Δs_{i}

**Problem 21**: let X and Y be linear spaces (vector spaces) of the same field F, and α^{1}, α^{2} 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(α ^{1}x_{1} + α^{2}x_{2}) = α^{1}T(x_{1}) + α^{2}T(x_{2})**

**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 V_{2}(R) into V_{2}(R) a linear transformation?
**Ans: let u _{1} = (x_{1}, y_{1});
**

and u_{2} = (x_{2}, y_{2}) in the domain of T. Let a , b be scalars in the real field R.

au_{1} = a(x_{1}, y_{1}); bu_{2} = b(x_{2}, y_{2})

T(au_{1} + bu_{2}) = (ax_{1} + bx_{2} - ay_{1} - by_{2})

aT(u_{1}) + bT(u_{2}) = (ax_{1} + bx_{2} - ay_{1} - by_{2})

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(au_{1} + bu_{2}) = A(au_{1} + bu_{2})

=aAu_{1} + bAu_{2} = aT(u_{1}) + bT(u_{2})

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 V_{n}(R) into V_{n}(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 R^{1} to R^{1} what is a general expression for T?
**Ans: T(x) = cx. **

**Problem 28**: is the transformation T(x,y) = (x, 3y) from R^{2} to R^{2} 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 = u_{1},...u_{n}; and v = v_{1},...v_{n}; in V_{n}(R) is defined as u.v = u_{1}v_{1} + u_{2}v_{2}...+ u_{n}v_{n} = |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 V_{3}(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 R^{2} to R^{2} (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 R ^{2}, θ degrees counterclockwise about the y axis. Yes T is a linear transformation. Solution will be clear after familiarization with matrices. **

Peter Oye Sagay