The Cauchy-Schwarz Inequality Let u and v be vectors in R 2
.
We wish to prove that -> (u · v)^ 2 ≤ |u|^
2 |v|^2 .
This inequality is called the Cauchy-Schwarz inequality and is
one of the most important inequalities in linear algebra.
One way to do this to use the angle relation of the dot product
(do it!). Another way is a bit longer, but can be considered an
application of optimization. First, assume that the...
show that for any two vectors u and v in an inner product
space
||u+v||^2+||u-v||^2=2(||u||^2+||v||^2)
give a geometric interpretation of this result fot he vector
space R^2
All vectors are in R^ n. Prove the following statements.
a) v·v=||v||2
b) If ||u||2 + ||v||2 = ||u + v||2, then u and v are
orthogonal.
c) (Schwarz inequality) |v · w| ≤ ||v||||w||.
Prove
1. For each u ∈ R n there is a v ∈ R n such that u + v= 0
2. For all u, v ∈ R n and a ∈ R, a(u + v) = au + av
3. For all u ∈ R n and a, b ∈ R, (a + b)u = au + bu
4. For all u ∈ R n , 1u=u
1. For this question, we define the following vectors: u = (1,
2), v = (−2, 3).
(a) Sketch following vectors on the same set of axes. Make sure
to label your axes with a scale. i. 2u ii. −v iii. u + 2v iv. A
unit vector which is parallel to v
(b) Let w be the vector satisfying u + v + w = 0 (0 is the zero
vector). Draw a diagram showing the geometric relationship between...
(1) Suppose that V is a vector space and that S = {u,v} is a set
of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so
thatT is another set of two vectors in V ). (a) Show that if S is
linearly independent in V then T is also independent. (Hint:
suppose that there is a linear combination of elements of T that is
equal to 0. Then ....). (b) Show that if S generates V...
1) Determine the angle between vectors:
U = <2, -3, 4> and V= <-1, 3, -2>
2) determine the distance between line and point
P: -2x+3y-4z =2
L: 3x – 5y+z =1
3) Determine the distance between the line L and the point A
given by
L; (x-1)/2 = (y+2)/5 = (z-3)/4 and A (1, -1,1)
4) Find an equation of the line given by the points A, B and
C.
A (2, -1,0), B (-2,4,-1) and C ( 3,-4,1)...
1. Consider the three vectors, u and v are 10 degrees above and
below the x-axis respectively, and ||u|| = 1, ||v|| = 2, and ||w||
= 3. Arrange the dot products taken among these vectors from least
to greatest
2. Let A be a 4 x 6 matrix. Find the elimination matrix E that
corresponds with the row operation "switch rows 1 and 3, and scale
row 4 by a factor of 6.
3. Find the formula for the...