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 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 = (u1,u2,u3) and the vector v = (v1,v2,v3) 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 = (u2v3 - u3v2,u3v1 - u1v3,u1v2 - u2v1)

Problem 47: Prove that in a vector space V, au = 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 R4. 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 R3: (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

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Peter Oye Sagay

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