What is Linear combinations: Definition and 61 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.

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  1. C

    One set v is a linear combination of u. Prove u is linearly dependent

    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...
  2. U

    I Limit of limits of linear combinations of indicator functions

    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...
  3. karush

    MHB Matrices.......whose null space consists all linear combinations

    $ v=\left[\begin{array}{r} -3\\-4\\-5\\4\\-1 \end{array}\right] w=\left[\begin{array}{r} -2\\0 \\1 \\4 \\-1 \end{array}\right] x=\left[\begin{array}{r} 2\\3 \\4 \\-5 \\0 \end{array}\right] y=\left[\begin{array}{r} -2\\1 \\0 \\-2 \\7 \end{array}\right] z=\left[\begin{array}{r} -1\\0 \\2 \\-3...
  4. P

    I Closure in the subspace of linear combinations of vectors

    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...
  5. J

    I Finding a linear combination to enter a sphere

    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...
  6. A

    I Linear combinations of atomic orbitals

    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...
  7. fisher garry

    I Bivariate normal distribution from normal linear combination

    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...
  8. S

    B Exploring Linear Combinations in Quantum Mechanics

    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...
  9. G

    I Quantum Superposition, Linear Combinations and Basis

    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...
  10. M

    Wrong answer on Linear Algebra and Its Applications 4th Ed.

    Homework Statement The Attempt at a Solution \left[ \begin{array}{cccc} 1 & 0 & 5 & 2 \\ -2 & 1 & -6 & -1 \\ 0 & 2 & 8 & 6 \end{array} \right] \sim \left[ \begin{array}{cccc} 1 & 0 & 5 & 2 \\ 0 & 1 & 4 & 3 \\ 0 & 0 & 0 & 0 \end{array} \right] From the RREF it is...
  11. H

    A Identical Fermions in identical linear combinations

    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?
  12. H

    B Mixed states V superposition V linear combinations?

    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...
  13. Destroxia

    Linear Combinations in 2-space

    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...
  14. kostoglotov

    Column space and nullspace relationship?

    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...
  15. B

    MHB Can a Matrix be Written as a Linear Combination of Another Matrix's Columns?

    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...
  16. C

    Cartesian to polar unit vectors + Linear Combination

    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]...
  17. _N3WTON_

    Linear Combinations and Span (Concept Question)

    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...
  18. I

    MHB Spanning Spaces with Linear Combinations

    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...
  19. A

    Linear combinations of 3 non-parallel vectors in 2D Euc.-space

    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}...
  20. U

    Sampling- linear combinations

    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...
  21. M

    Linear combinations of non-eigenfunctions to create eigenfunctions

    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...
  22. B

    Understanding Linear Combinations of Vectors

    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...
  23. C

    MHB Describe Geometrically (line, plane, or all of R3) all linear combinations 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...
  24. K

    Questions about Linear Combinations of Random Variables

    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...
  25. alane1994

    MHB Mechanical Vibrations - Linear Combinations

    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...
  26. B

    Linear combination of linear combinations?

    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...
  27. K

    MHB Linear Combinations and description geometrically

    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...
  28. T

    Writing orthogonal vectors as linear combinations

    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...
  29. M

    Linear dependence and independence; linear combinations

    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?
  30. P

    Analyzing Linear Combinations: v1, v2 and v3

    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...
  31. N

    Linear Systems: Linear combinations

    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...
  32. G

    Find normalised linear combinations that are orthogonal

    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...
  33. H

    Linear Combinations: Solving for 4th Vector with 3 Vectors

    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...
  34. S

    Linear combinations euclidean algoritm extended

    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...
  35. S

    Linear Independence: Writing vectors as linear combinations

    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...
  36. J

    Vectors: Linear Combinations and Parallelism

    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...
  37. M

    Testing for linear combinations using matrices instead 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...
  38. Z

    Nondegenerate Eignefunctions as Linear Combinations

    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...
  39. M

    Calculating Uncertainties with linear combinations

    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...
  40. P

    Linear Combinations: Will Two Always Produce b=(0,1)?

    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)
  41. P

    Linear Algebra linear combinations help

    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...
  42. D

    Linear Combinations of Trig Functions - Finding Roots

    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...
  43. P

    Solving Linear Combinations: (1,2,3)

    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...
  44. S

    Simplified Heisenberg Hamiltonian; Linear combinations of basis states

    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...
  45. M

    Linear Algebra: Linear Independence and writing Matrices as linear combinations

    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...
  46. L

    Linear Algebra: Linear Combinations

    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...
  47. R

    Linear Combinations of Dependent Vectors

    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...
  48. E

    Mean and standard deviation for linear combinations

    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...
  49. A

    Question on linear combinations of sines and cosine (complex analysis)

    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...
  50. A

    Linear combinations of structure constants

    I have two Lie algebras with structure constants f^{a}_{bc} and g^{a}_{bc}, the number of generators being the same (as will become clear). Due to a particular symmetry/construct, I have that the system needs to be valid under g \to af + bg (and a similar transform for g), which leads...
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