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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 11**: what is the dimension of the following vector spaces:

(i) V_{2}(R)

(ii) V_{3}(R)
**Ans: (i) the dimension of V _{2}(R) is 2
(ii) the dimension of V_{3}(R) is 3.
Dimensions 1, 2 and 3 are the dimensions we perceive in the real world. However, mathematicians work with higher dimensions in advanced abstract analysis. **

**Problem 12**: i = (1,0,0), j = (0,1,0), and k = (0,0,1) are basis for the vector space V_{3}(R). Why?
**Ans: i, j and k are basis for V _{3}(R) because any vector v = (x, y, z)
can be written in the form xi + yj + zk. **

**Problem 13**: define a family of points in a linear space X over the field F.
**Ans: Let S be an index set and f a one-one mapping from S into X such that f(s) = x _{s}, the points (x_{s}) for all s in S, is a family of points in the linear space X. **

**Problem 14 **: define a subfamily of the family (x_{s}) of points in the linear space X over the field F.
**Ans: Let S' be a subset of the index set S that generated the family of points (x _{s}). If f, a one-one mapping from S into X is restricted to only the elements of S' such that f(s') = (x_{s'}), then the points (x_{s'}) for all s' in S' is a subfamily of the points (x_{s}) in the linear space X over the field F. The subfamily is finite if S' has a finite number of elements. **

**Problem 15 **: how is the linear independence of the family (x_{s}) of points in the linear space X determined?
**Ans: Let α ^{s} be a set of scalars in the field F, and s an arbitrary member of the subset S' of the index set S. If for each subfamily (x_{s})_{s in S'}, ∑α^{s}x_{s} = 0 implies α^{s} = 0 for all s in S', then the family (x_{s})_{s in S} of points in X is said to be linearly independent. A family of points in X is linearly dependent if it is not linearly independent.
**

**Problem 16**: show that the basis i = (1,0,0), j = (0,1,0), and k = (0,0,1) in the space V_{3}(R), are linearly independent.
**Ans: let α ^{1}, α^{2}, α^{3} be the scalars. Then ∑α^{s}x_{s} = α^{1}(1,0,0) + α^{2}(0,1,0) + α^{3}(0,0,1) = (0, 0,0).
**

= (α^{1}, α^{2}, α^{3}) = (0, 0, 0)

This implies α^{1} = α^{2} = α^{3} = 0. Therefore i, j, and k are linearly independent.

i, j, and k are called unit vectors of the vector space V_{3}(R)

In general, the sum ∑α^{s}x_{s} is called a linear combination of the vectors and α^{s} is called the coefficient of the vector x_{s}.

**Problem 17**: if W is a subspace of the linear space X, explain the statement: W spans X.
**Ans: W spans X if every element of X can be expressed as a linear combination of a finite number of elements of W. the vectors i, j, k of problem 16 span the vector space V _{3}(R). **

**Problem 18**: let x_{s} be the range of a linearly independent family (x_{s})_{s in S} of points in a linear space X (S is an index set)and let if x_{s} span X. Therefore; x_{s} is a Hamel basis. The number of elements in x_{s} are infinite. True or False?
**Ans: False. **

**Problem 19**: the number of elements in a Hamel basis is called its cardinal number. This cardinal number is called the dimension of the space. If (e_{i}, ..., e_{n}) is the Hamel basis of the space X, what is the dimension of X.
**Ans: try it. **

**Problem 20**: (e_{i}, ..., e_{n}) is a Hamel basis for the space X; (e_{j}, ..., e_{m}) is also a Hamel basis for the space X. If n = 7 what is (m + n) mod 14.
**Ans: try it. hint: any two Hamel bases for a linear space has the same cardinal number. **

Peter Oye Sagay