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⇒Homework Statement
[/B]
Calculate ##S + T## and determine if the sum is direct for the following subspaces of ##\mathbf R^3##
a) ## S = \{(x,y,z) \in \mathbf R^3 : x =z\}##
## T = \{(x,y,z) \in R^3 : z = 0\}##
b) ## S = \{(x,y,z) \in \mathbf R^3 : x = y\}##
## T = \{(x,y,z) \in \mathbf R^3 ...
Homework Statement
Let ##V## be the vector space of the sequences which take real values. Prove whether or not the following subsets ##W \in V## are subspaces of ##(V, +, \cdot)##
a) ## W = \{(a_n) \in V : \sum_{n=1}^\infty |a_n| < \infty\} ##
b) ## W = \{(a_n) \in V : \lim_{n\to \infty} a_n...
Homework Statement
Let V = RR be the vector space of the pointwise functions from R to R. Determine whether or not the following subsets W contained in V are subspaces of V.
Homework Equations
W = {f ∈ V : f(1) = 1}
W = {f ∈ V: f(1) = 0}
W = {f ∈ V : ∃f ''(0)}
W = {f ∈ V: ∃f ''(x) ∀x ∈ R}
The...
I wanted to go through Calculus and then Linear Algebra following either of two paths:
a) Keisler's Infinitesmal approach>>>Nitecki Deconstructing Calculus>>>Nitecki Calculus in 3D>>>Freidberg's Linear Algebra
OR
b) Simmons Calculus with analytic geometry>>>Apostol Vol 1>>>>Apostol Vol...
Dear Fellows,
I have recently completed the study of Stewart's calculus. Next, I want to read Linear Algebra.
I have bought Sheldon Axler's "Linear Algebra done right" textbook. I want to know if my knowledge of calculus is enough to tackle this book or should I first...
Homework Statement
Find the distance from point P (1,7,3) to the line
(x,y,z) = (-2,1,4) + s(1,-3,4),
s is a free variable
Homework Equations
projnQP = ( QP⋅n/(lengthQP)(lengthn) )(n)
The Attempt at a Solution
I'm not quite sure about how to find the normal (n) here, but if I make s=0, I'm...
Homework Statement
From Linear Algebra and Its Applications, 5th Edition, David Lay
Chapter 4, Section 1, Question 32
Let H and K be subspaces of a vector space V. The intersection of H and K is the set of v in V that belong to both H and K. Show that H ∩ K is a subspace of V. (See figure.)...
Homework Statement
If I have two eigenfunctions of some operator, that are linearly indepdendent e.g ##F(x) , G(x)+16F(x) ## and ##F(x)## has eigenvalue ##0##, does this mean that ## G(x) ## must itself be an eigenfunction?
I thought for sure yes, but the way I particular question I just...
Homework Statement
Suppose that we have a system consisting of two interconnected tanks, each containing a brine solution. Tank A contains
x(t) pounds of salt in 200 gallons of brine, and tank B contains y(t) pounds of salt in 300 gallons of brine. The mixture in each tank is kept uniform by...
Homework Statement
Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K.
Homework Equations
f1, ..., fn are field morphisms => Ker (fi) = 0 (injective)
The Attempt at a Solution
I...
Homework Statement
Problem- Determine if the set of all function y(t) which have period 2pi forms a vector space under operations of function addition and multiplication of a function by a constant.
What I know- So I know this involves sin, cos, sec, and csc. Also I know that a vector space...
Hi physicsforums,
I am an undergrad currently taking an upper-division course in Quantum Mechanics and we have begun studying L^2 space, state vectors, bra-ket notation, and operators, etc.
I have a few questions about the relationship between L^2, the space of square-integrable complex-valued...
So, I recently came across this example: let us "define" a function as ƒ(x)=-x3-2x -3. If given a matrix A, compute ƒ(A). The soution proceedes in finding -A3-2A-3I where I is the multiplicative identity matrix.
Now , I understand that you can't add a scalar and a matrix, so the way I see it is...
Homework Statement
Suppose that A is a 3 x n matrix whose columns span R3. Explain how to construct an n x 3 matrix D such that AD = I3.
"Theorem 4"
For a matrix A of size m x n, the following statements are equivalent, that is either all true or all false:
a. For each b in Rm, Ax = b has a...
Given an ##N## dimensional binary vector ##\mathbf{v}## whose conversion to decimal is equal to ##j##, is there a way to linearly map the vector ##\mathbf{v}## to an ##{2^N}## dimensional binary vector ##\mathbf{e}## whose ##(j+1)##-th element is equal to ##1## (assuming the index starts...
The question comes out of a corollary of this theorem:
Let B be a symmetric bilinear form on a vector space, V, over a field \mathbb{F}= \mathbb{R} or \mathbb{F}= \mathbb{C}. Then there exists a basis v_{1},\dots, v_{n} such that B(v_{i},v_{j}) = 0 for i\neq j and such that for all...
Homework Statement
Find the Fourier series of the function ##f## given by ##f(x) = 1##, ##|x| \geq \frac{\pi}{2}## and ##f(x) = 0##, ##|x| \leq \frac{\pi}{2}## over the interval ##[-\pi, \pi]##.
Homework Equations
From my lecture notes, the Fourier series is
##f(t) = \frac{a_0}{2}*1 +...
<Moderator's note: Moved from a technical forum and thus no template.>
Hello I am given the following problem to solve.
Identify the quadratic form given by ##-5x^2 + y^2 - z^2 + 4xy + 6xz = 5##.
Finally, plot it.
I cannot seem to understand what I have to do. The textbook chapter on...
Homework Statement
How to prove ##max\{0, \rho(\sigma)+\rho(\tau)-m\}\leq \rho(\tau\sigma)\leq min\{\rho(\tau), \rho(\sigma)\}##?
Homework Equations
Let ##\sigma:U\rightarrow V## and ##\tau:V\rightarrow W## such that ##dimU=n##, ##dimV=m##. Define ##v(\tau)## to be the nullity of ##\tau##...
Homework Statement
Homework Equations
I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...
A problem that I have to solve for my Linear Algebra course is the following
We are supposed to use Mathematica.
What I have done is that I first checked that A is symmetric, i.e. that ##A = A^T##. Which is obvious.
Next I computed the eigenvalues for A. The characteristic polynomial is...
I am trying to understand the geometric intuition of the above equation. ##\rho(\tau)## represents the rank of the linear transformation ##\tau## and likewise for ##\rho(\tau\sigma)##. ##Im(\sigma)## means the image of the linear transformation ##\sigma## and lastly, ##K(\tau)## is the kernel of...
Homework Statement
Let ##W_1=\langle (1,2,3,6),(4,-1,3,6)(5,1,6,12))\rangle## and ##W_2=\langle (1,-1,1,1),(2,-1,4,5)\rangle## be subspaces of ##\Bbb{R}^4##. Find the bases for ##W_1\cap W_2## and ##W_1+W_2##.
Homework Equations...