What is Linear combinations: Definition and 62 Discussions
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics.
Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
Hi, as in this thread Are maxwells equations linearly dependent I would like to better understand from a mathematical point of view the interdependence of Maxwell's equations.
Maxwell's equations are solved assuming as given/fixed the charge density ##\rho## and the current density ##J## as...
Hi Everybody,
I am having some difficulties on the prove this problem.
I picked a nice example when I was trying to think about the proof.
Let ##s=3## and ##t=2##. Then ##u1=c1v1+c2v2, u2=c3v1+c4v2, u3=c5v1+c6v2##. Then a linear combination of u: ##K1u1+K2u2+K3u3=0##. I grouped both linear...
I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##.
Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...
This is the exact definition and I've summarized it, as I understand it above. Why is it, that for elements in the third subspace, closure will be lost? Wouldn't you still get another vector (when you add two vectors in that subspace), that's still a linear combination of the vectors in the...
Let's say we have n vectors in ℝ3. And say we have defined a subspace inside ℝ3 in the form of a sphere with radius r, and the center of the spheare is at P, where P is a vector in ℝ3.
What methods exists to find any linear combination of the n vectors, so that the sum of all of them, lies...
So I've been looking at covalent bonds and come across the approx you can do of the molecular orbital for ##H^+_2## by just summing two 1s orbitals, the method is called the linear combinations of atomic orbitals, and you get what is below which I believe is exact in this case since the 1s...
I can't prove this proposition. I have however managed to prove that the linear combinations of the independent normal rv's are also normal by looking at it's mgf
$$E(e^{X_1+X_2+...+X_n})=E(e^{X_1})E(e^{X_2})...E(e^{X_n})$$
The mgf of a normal distribution is $$e^{\mu t}e^{\frac{t^2...
Hi, I read that linear combinations of a state, Psi, can be as:
\begin{equation}
\Psi = \alpha \psi + \beta \psi
\end{equation}
where ##\alpha## and ##\beta## are arbitrary constants.
Can however this be a valid linear combination?\begin{equation}
\Psi = \alpha \psi \times \beta \psi...
Hello, just a quick question.
I am aware that a a state in a space can be written as a linear combination of the basis kets of that space
ψ = ∑ai[ψi]
where ai are coefficients and [ψi] are the basis vectors.
I was just wondering is this a linear superposition of states or just a linear...
According to Pauli principle the two fermions can not occupy one state of a Hamiltonian. Can the two fermions occupy a state which is linear recombination of two states of the Hamiltonian?
Can someone explain the difference using concrete examples. I will attempt to explain my current understanding by example;
A H atom has different energy levels which can be exactly described by algebraic functions with quantum numbers n, l etc.
An electron can be excited from say the ground...
Homework Statement
In the xy-plane mark all nine of these linear combinations:
## α \lbrack 2, 1 \rbrack + β \lbrack 0, 1 \rbrack## with c = 0, 1, 2 and d = 0, 1, 2 Homework Equations
ANSWER:
The nine combinations will lie on a lattice. If we took all whole numbers c and d, the lattice...
I have just been studying Nullspaces...
I want to make the following summary, will it be correct?
C(A) is all possible linear combinations of the pivot columns of A.
N(A) is all possible linear combinations of the free columns of A (if any exist).
edit: I have a feeling these are...
So i have the following:
[a b; c d] = [e f ; g h] * [p q ; r s]
I have to show that the if the original matrices are written as A = EP then the columns of A are linear combinations of E.
I was able to prove [a;c] = p[e;g] + r[f;h] and the same for [b;d] but i don't know where to go from here...
I've been trying to solve this question all day. If somebody could point me in the right direction I would really appreciate it!
(ii) A particle’s motion is described by the following position vector r(t) = 4txˆ + (10t − t)ˆy Determine the polar coordinate unit vectors ˆr and ˆθ for r. [4]...
Homework Statement
Let A be an m \hspace{1 mm} x \hspace{1 mm} n matrix, and let \vec{b} be a vector in \mathbb{R}^{m} . Suppose that \vec{b} is a linear combination of the columns of A. Then the columns of A span \mathbb{R}^{m}
Homework EquationsThe Attempt at a Solution
I said that...
Hello!
Christina, thank you for this thread. I also have the same issue.
I like Serena, I would be grateful for your further help.
If given vectors are (1,2,3) and (3,6,9) that have v3 = v1+v2, does it mean that the whole V vector or W vector lies only on XOY plane (two dimensional) and OZ is...
Homework Statement
Suppose \mathbf{u,v,w} \in {\rm I\!R}^2 are noncollinear points, and let \mathbf{x} \in {\rm I\!R}^2.
Show that we can write \mathbf{x} uniquely in the form \mathbf{x} = r\mathbf{u} + s\mathbf{v} + t\mathbf{w}, where r + s + t = 1.
Homework Equations
Suppose \mathbf{a,b}...
The scores X1 and X2 in papers 1 and 2 of an examination are normally distributed with means 24.3 and 31.2 respectively and standard deviations 3.5 and 3.1 respectively The final mark for each candidate is found by calculating 2X1+1.5X2. Find the probability that a random sample of 8candiates...
Homework Statement
Consider the Parity Operator, P', of a single variable function, defined as P'ψ(x)=P'(-x).
Let ψ1=(1+x)/(1+x^2) and ψ2=(1+x)/(1+x^2). I have already shown that these are not eigenfunctions of P'.
The question asks me to find what linear combinations, Θ=aψ1+bψ2 are...
Hello Everyone,
Pardon me if the following is incoherent. From what I understood of what my professor said, he was basically saying that when a vector can be written as a linear combination of some vectors in a span, this means, geometrically, that the vector is in the plane that the span of...
I know that someone posted this before, however I could not respond to that thread. I have not taken a Math course in several years and although I can do basic math and algebra, linear algebra is already seeming to be quite difficult. Basically with the title(subject line) the linear...
Homework Statement
Homework Equations
Y=1/2*(X1-X3)^2+1/14*(X2+2X4-3X5)^2The Attempt at a Solution
For (a) part, I have only learned to find the moment-generating function of Y, but not finding the p.d.f.
Moreover, the examples I have seen only involves random variables Xi to the power 1, but...
The title may be incorrect, I named this after the section of my book in which this is located.
My problem is as follows.
Determine \(\omega_0\), R, and \(\delta\) so as to write the given expression in the form
\(u=R\cos(\omega_0 t-\delta)\)
\(\color{blue}{u=4\cos(3t)-2\sin(3t)},~\text{My...
When the book says "Members of [[S]] are linear combinations of linear combinations of members of S". [S] basically means the span of the members in subspace S. Since
[S] = {c1s1 +... + cnsn|c1...cnεR and s1...snεS} what does [[S]] mean? does it mean a linear combination of atleast one linear...
Ok give me a break, this is my first lesson in my new linear algebra book. Seems fairly straightforward but a little befuddled as to whether I am doing this. The question states "Describe geometrically (line, plane, or all of R^3) all linear combinations of..."
Then I have a matrix v = [1 2 3] w...
Hello,
Quick question, not really homework but more of a general inquiry. Take three vectors: a,b and c such that a and c are orthogonal. Is it possible to write c as a linear combination of a and b such that:
c = ma + nb where m,n are scalars.
I was thinking not at first glance but...
I cannot visualize the geometry for either of these ideas. Is it the case that two vectors can be linearly independent or dependent of each other? In which case, what is the dependency or independency based on? What are these two vectors independent or dependent of with respect to each other?
Lets say you have 3 vectors v1, v2,v3. They form a 3x3 matrix.
Let's say you're asked if v3 is a linear combo of the other two vectors.
Rref of the matrix gives
1 0 1
0 1 0
0 0 0
The definition of a Linear combo is v3=c1v1+c2v2 where c1 and c2 are scalars.
Okay do this is where...
Homework Statement
Determine if b is a linear combination of a1, a2, and a3.
a1= [1,-2, 0]
a2 = [0, 1, 2]
a3 = [5, -6 8]
b= [2, -1, 6]
The Attempt at a Solution
Alright, well I used an augmented matrix to solve the problem, and after reducing it completely, the matrix looked...
Homework Statement
I'm a little weary of posting this in this forum. If I post it in the math section it will be answered in about 30 min whereas here it might take about 5 hours, but we'll see.
Homework Equations
The Attempt at a Solution
Number one, I'm not exactly sure how they get from...
Hi
If i have 3 4x1 matrices and i want to check if i can express a 4th matrix as the linear combination of the first 3.
my 3 vectors:
1 7 -2
4 10 1
2 -4 5
-3 -1 -4
can this vector be expressed a linear combination of the first...
Homework Statement
I have questions along the line of
Use the Euclidean Algorithm to find d= gcd(a,b) and x, y \in Z with d= ax +by
Homework Equations
The Attempt at a Solution
Ok so I use the euclidean algorithm as I know it on say gcd (83,36), by minusing of the the...
Forgive me for not writing in latex, but I searched this site for 10 minutes looking for a latex reference and could not find anything on matrices. Also, excuse for the excessive amount of info.
Homework Statement
Determine whether this list of 3 polynomials in P1:
p1 = 1+3x
p2 = 1+2x...
Homework Statement
The question states: "find all scalars c, if any exist such that given statement is true."
It also suggests trying to do without pencil and paper.
a) vector [c2, c3, c4] is parallel to [1,-2,4] with same direction.
b) vector [13, -15] is a linear combination of vectors...
I want to see if the matrix w = (1,0;0,1) is a linear combination of the matrices
v1 = (1,2;-2,1) and v2 = (3,2;-1,1) where ; denotes a new line in the matrix.
I know for example if w and v were 1xn matrices i.e vectors such as w = [1,1,1]
v1= [2,-1,3] v2=[1,1,2] then i setup a matrix with...
It is easily shown that two eigenfunctions with the same eigenvalues can be combined in a linear combination so that the linear combination is itself an eigenfunction. But what if the two eigenvalues are not the same? Can you still find a linear combination of the two functions that is an...
Homework Statement
I am having trouble determining the error for a set of linear equations that represent a simple circuit with two voltage sources. I have found two possible uncertainties by solving using substitution, detailed below.
The circuit is shown below...
Will there always be two different combinations that produce b=(0,1) of three vectors: u, v, and w?
I'm pretty certain that the answer is no, but am I right in saying that with three vectors, assuming they are not all parallel, will always have at least one combination that produces (0,1)
The linear combinations of v=(a,b) and w=(c,d) fill the plane unless _____.
Find four vectors u, v, w, z with four components each so that their combinations cu+dv+ew+fz produce all vectors (b1, b2, b3, b4) in four dimensional space.
I think that the first part of the answer, that fills the...
Hi there
I was wondering if there is a simple way to solve for the roots of a complicated summation of trig functions that can't be combined with any simple identities.
I have an equation of the form:
0 = sin(8x-arctan(4/3))+3.2sin(16x+pi/2)
where the two sines have different amplitudes...
Homework Statement
Write the vector (1,2,3) as a linear combination of the vectors (1,0,1), (1,0,-1), and (0,1,1).
The attempt at a solution
(1,2,3) = C1(1,0,1) + C2(1,0,-1) + C3(0,1,1)
The matrix for this is:
1...1...0...1
0...0...1...2
1...-1...1...3
I reduced it to the...
So, I'm doing some undergraduate research in quantum spin systems, looking at the ground states of the Heisenberg Hamiltonian, H=\sum{J_{ij}\textbf{S}_{i}\textbf{S}_{j}}. But I think I have a critical misunderstanding of some fundamental quantum mechanics concepts. (I'm a math major, only had...
Homework Statement
If linearly dependent, write one matrix as a linear combination of the rest.
\left[\begin{array}{cc} 1&1 \\ 2&1 \end{array}\right] \left[\begin{array}{cc} 1&0 \\ 0&2 \end{array}\right] \left[\begin{array}{cc} 0&3 \\ 2&1 \end{array}\right] \left[\begin{array}{cc} 4&6 \\ 8&6...
Homework Statement
Let V = {f: \mathbb {R}\rightarrow\mathbb {R}} be the vector space of functions. Are f1 = ex, f2 = e-x (both \in V) linearly independent?
Homework Equations
0 = aex + be-x Does a = b = 0?
The Attempt at a Solution
My first try, I put a = e-x and b = -ex. He...
Homework Statement
If (u,v,w) is a family of linearly dependent vectors in vector space V and vector x is in the span of (u,v,w), then x=αu+βv+γw has infinitely-many choices for α,β, and γ.
Homework Equations
If (u,v,w) is linearly dependent, then there exists an α, β, and γ, not all equal...
Homework Statement
Data set in X with mean xbar = 100 and standard deviation Sx = 10
Find ybar and Sy for 2(Yi-5)/10 + 7
Homework Equations
The Attempt at a Solution
All the problems I have seen are in the form yi = axi + b in which case the mean ybar = a(xbar) + b and...
I have a question on complex analysis. Given a differential equation,
\dfrac{d^2 \psi}{dx^2} + k ^2 \psi = 0
we know that the general solution (before imposing any boundary conditions) is,
\psi (x) = A cos(kx)+B sin(kx).
Now here's something I don't quite understand. The solution...