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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 41**: what is a normed linear space?
**Ans: let X be a linear space over the field F where F is either the real field R or the complex field C.
A norm on X is a real-valued function whose value at x is denoted by ||x||, satisfying the following conditions for all x, y, in X and &lpha; in F:
(i)||x|| > 0 if x ≠ 0
(ii)||αx|| = |α|||x||
(iii)||x + y|| ≤ ||x|| + ||y||
A linear space X with a norm defined on it is called a normed linear space. **

**Problem 42**: Consider a vector space with origin at O in which the vector **u** is represented by the arrow from O to P and the vector **v** is represented by the arrow from O to Q. The projection **v** on **u** is represented by the arrow from O to N, where N is the foot of the perpendicular drawn from Q to the line of **u**. Show that this projection is given by:
(**u.v**)**u**/|**u**|^{2}.
**Ans: The projection of v on u = (u/|v|)(|v|cosθ), where θ is the angle between u and v.
**

u.v = |u||v|cosθ (dot or scalar product). So, cosθ = (u.v)/(|u|(|v|). Use this expression for cosθ to replace the cosθ in

(u/|v|)(|v|cosθ) to get the desired result.

**Problem 43**: Determine the projection of (1,-2,1) on (3,1,-4).
**Ans: (-9/26,-3/26,6/13) **

**Problem 44 **: Find an equation of the line through the two points (3,-5,7) and (-2,1,4).
**Ans: The vector (a,b.c) = (-2-3, 1+5,4-7) = (-5,6,-3) specifies the direction of the line. Therefore, equation for the line is:
(x,y,z) = (3,-5,7) + t(-5,6,-3), -∞ < t < ∞ **

**Problem 45 **: Find the equation of the plane containing the two lines (x,y,z) = (1,-3,4) + t(2,5,-1) and (x,y,z) = (1,-3,4) + s(0,7,-4).
**Ans: equation of the plane (x,y,z) = (1,-3,4) + s(0,7,-4) + t(2,5,-1), -∞ < s < ∞, -∞ < t < ∞ **

**Problem 46**: the vector **u** = (u_{1},u_{2},u_{3}) and the vector **v** = (v_{1},v_{2},v_{3}) are nonzero and nonparallel. The vector **w** is orthogonal to both **u** and **v**. Determine **w**.
**Ans: w is the vector product of u and v thus:
**

w = u x v = (u_{2}v_{3} - u_{3}v_{2},u_{3}v_{1} - u_{1}v_{3},u_{1}v_{2} - u_{2}v_{1})

**Problem 47**: Prove that in a vector space V, a**u** = **0** implies that a = 0. or **u** = **0**, or both.
**Ans: try it. **

**Problem 48**: The vectors (1,1,1,1),(1,0,1,0), (0,1,0,1),(1,-1,1-1) span a subspace of R^{4}. Confirm that the vector (4,-2,4,-2) is in that subspace.
**Ans: (4,-2,4,-2) = 4(1,0,1,0) - 2(0,1,0,1). **

**Problem 49**: Construct orthonormal basis from the following vectors in R^{3}: (1,0,1), (1,-1,1), (0,1,1).
**Ans: try it. **

**Problem 50**: Find the projection of (1,2,3) on the plane given implicitly by x + y +z = 0
**Ans: try it **

Peter Oye Sagay