Page 98 - 35Linear Algebra
P. 98
98 Vectors in Space, n-Vectors
(∞) What is the limit as n → ∞ of the angle between the diagonal of
n
the unit (hyper)-cube in R and one of the coordinate axes?
cos θ sin θ x
3. Consider the matrix M = and the vector X = .
− sin θ cos θ y
2
(a) Sketch X and MX in R for several values of X and θ.
||MX||
(b) Compute for arbitrary values of X and θ.
||X||
(c) Explain your result for (b) and describe the action of M geomet-
rically.
n
4. (Lorentzian Strangeness). For this problem, consider R with the
Lorentzian inner product defined in example 53 above.
(a) Find a non-zero vector in two-dimensional Lorentzian space-time
with zero length.
(b) Find and sketch the collection of all vectors in two-dimensional
Lorentzian space-time with zero length.
(c) Find and sketch the collection of all vectors in three-dimensional
Lorentzian space-time with zero length.
(d) Replace the word “zero” with the word “one” in the previous two
parts.
The Story of Your Life
5. Create a system of equations whose solution set is a 99 dimensional
101
hyperplane in R .
3
6. Recall that a plane in R can be described by the equation
x
n · = n · p
y
z
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