Let u and v be two integers and let us assume u^2 + uv +v^2 is...
Let u and v be two integers and let us assume u^2 + uv +v^2 is
divisible by 9. Show that then u and v are divisible by 3. (please
do this by contrapositive).
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
Let A ∈ L(U, V ) and B ∈ L(V, W). Assume that V is
finite-dimensional.
Let X be a 3 × 5 matrix and Y be a 5 × 2 matrix. What are the
largest and smallest possible ranks of X, Y, and XY? Give examples
of the matrix to support your answers
Let X ∈ L(U, V ) and Y ∈ L(V, W). You may assume that V is
finite-dimensional.
1)Prove that dim(range Y) ≤ min(dim V, dim W). Explain the
corresponding result for matrices in terms of rank
2) If dim(range Y) = dim V, what can you conclude of Y? Give
some explanation
3) If dim(range Y) = dim W, what can you conclude of Y? Give
some explanation
Let U and V be vector spaces, and let L(V,U) be the set of all
linear transformations from V to U. Let T_1 and T_2 be in
L(V,U),v be in V, and x a real number. Define
vector addition in L(V,U) by
(T_1+T_2)(v)=T_1(v)+T_2(v)
, and define scalar multiplication of linear maps as
(xT)(v)=xT(v). Show that under
these operations, L(V,U) is a vector space.
Let Zt = U sin(2*pi*t) + V cos(2*pi*t), where U and V are
independent random variables, each with
mean 0 and variance 1.
(a) Is Zt strictly stationary?
(b) Is Zt weakly stationary?
True/False Question: If sppan{u,v}=W where u not equal to v.
Then dim(W)=2.
Answer: False
Reasoning let u=1, v=2 then span(1,2}=R but dim(R)=1 not 2.
I know the answer is false. please tell me whether my reasoning
is correct.
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...