What is Linear independence: Definition and 175 Discussions

In the theory of vector spaces, a set of vectors is said to be linearly dependent if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be linearly independent. These concepts are central to the definition of dimension.A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.

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  1. Z

    Linearly independent functions with identically zero Wronskian

    My question will be about item (c). Part (a) Note that for ##x\geq 0## we have ##f(x)=g(x)##. For ##x<0## we have ##f(x)=-g(x)##. Since ##f## is a constant times ##g## then one column of the matrix in the Wronskian is a constant times the other column. Thus, the Wronskian is zero, Note that...
  2. chwala

    I Determining linear indepedence/dependence of a set of functions

    Looking at the wronskian applications- came across this; Okay, i noted that one can also have this approach(just differentiate directly). Sharing just incase one has more insight. ##-18c \sin 2x -4k\cos x \sin x - 4k\sin x\cos x =0## ##-18c\sin 2x-2k\sin2x-2k\sin 2x=0## ##-18c\sin 2x =...
  3. Salmone

    I Linear independence of three vectors

    If I've got three vectors ##\vec{a}##, ##\vec{b}## and ##\vec{c}## and ##\vec{a}##, ##\vec{b}## are linearly independent and ##\vec{c}## is linearly independent from ##\vec{a}##, is ##\vec{c}## also linearly independent from ##\vec{b}##?
  4. isaacdl

    A Proving $g(u,v)≠0$ with Linear Independence

    I'm trying to prove that there exist always a vector w whose contraction with a lightlike vector u (g(u,u)=0) it's always different from zero: $g(u,v)≠0$I know how to do this with coordinates, but in a free cordinate scheme I'm totally lost. Any help? PD: Both vectors are linearly independent.
  5. C

    Linear independence of Coordinate vectors as columns & rows

    Summary:: x Question: Book's Answer: My attempt: The coordinate vectors of the matrices w.r.t to the standard basis of ## M_2(\mathbb{R}) ## are: ## \lbrack A \rbrack = \begin{bmatrix}1\\2\\-3\\4\\0\\1 \end{bmatrix} , \lbrack B \rbrack = \begin{bmatrix}1\\3\\-4\\6\\5\\4 \end{bmatrix}...
  6. F

    I Proving linear independence of two functions in a vector space

    Hello, I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...
  7. K

    I Trying to get a better understanding of the quotient V/U in linear algebra

    Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...
  8. brotherbobby

    Proving that the two given functions are linearly independent

    Summary:: I attach a picture of the given problem below, just before my attempt to solve it. We are required to show that ##\alpha_1 \varphi_1(t) + \alpha_2 \varphi_2(t) = 0## for some ##\alpha_1, \alpha_2 \in \mathbb{R}## is only possible when both ##\alpha_1, \alpha_2 = 0##. I don't know...
  9. GlassBones

    Isomorphisms preserve linear independence

    Homework Statement Let ##T:V \rightarrow W## be an ismorphism. Let ##\{v_1, ..., v_k\}## be a subset of V. Prove that ##\{v_1, ..., v_k\}## is a linearly independent set if and only if ##\{T(v_1), ... , T(v_2)\}## is a linearly independent set. Homework EquationsThe Attempt at a Solution...
  10. M

    A Numerics: Wronskian and linear independence

    Hi PF! I'm solving a differential eigen-value problem in weak form, so I have trial functions. If the Wronskian of trial functions is small but not zero, is linear independence an issue? I have analytic trial functions but am numerically integrating.
  11. hilbert2

    A Linearly independent function sets

    It is well known that the set of exponential functions ##f:\mathbb{R}\rightarrow \mathbb{R}_+ : f(x)=e^{-kx}##, with ##k\in\mathbb{R}## is linearly independent. So is the set of sine functions ##f:\mathbb{R}\rightarrow [-1,1]: f(x) = \sin kx##, with ##k\in\mathbb{R}_+##. What about...
  12. J

    MHB Linear Map, T^p(x)=0, Show Linear Independence

    Problem: Suppose V is a complex vector space of dimension n, and T is a linear map from V to V. Suppose $x \in V$, and p is a positive integer such that $T^p(x)=0$ but $T^{p-1}(x)\ne0$. Show that $x, Tx, T^2x, ... , T^{p-1}x$ are linearly independent.During class my professor said it was "a...
  13. Pushoam

    Checking the linear independence of elements of 2 X 2 matrices

    Homework Statement Homework Equations 3. The Attempt at a Solution [/B] ## |3 \rangle = |1 \rangle - 2 ~ |2 \rangle ## So, they are not linearly independent. One way to find the coefficients is : ## |3 \rangle = a~ |1 \rangle +b~ |2 \rangle ## ...(1) And solve (1) to get the values of a...
  14. M

    Linear algebra, field morphisms and linear independence

    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...
  15. L

    MHB Question about proof of the linear independence of a dual basis

    This is from Kreyszig's Introductory Functional Analysis Theorem 2.9-1. Let $X$ be an n-dimensional vector space and $E=\{e_1, \cdots, e_n \}$ a basis for $X$. Then $F = \{f_1, \cdots, f_n\}$ given by (6) is a basis for the algebraic dual $X^*$ of $X$, and $\text{dim}X^* = \text{dim}X=n$...
  16. LarryS

    I Measures of Linear Independence?

    My formal education in Linear Algebra was lacking, so I have been studying that subject lately, especially the subject of Linear Independence. I'm looking for functions that would qualify as measures of linear independence. Specifically, given a real-valued vector space V of finite dimension...
  17. Drakkith

    Linear Independence of Two Functions

    Homework Statement Use definition (1) to determine if the functions ##y_1## and ##y_2## are linearly dependent on the interval (0,1). ##y_1(t)=cos(t)sin(t)## ##y_2(t)=sin(t)## Homework Equations (1) A pair of functions is said to be linearly independent on the interval ##I## if and only if...
  18. cathal84

    I Proving a set is linearly independant

    I have two questions for you. Typically when trying to find out if a set of vectors is linearly independent i put the vectors into a matrix and do RREF and based on that i can tell if the set of vectors is linearly independent. If there is no zero rows in the RREF i can say that the vectors are...
  19. Mr Davis 97

    T/F Question of linear independence

    Homework Statement T/F: Let ##T: V \rightarrow W##. If ##\{v_1,v_2,...,v_k \}## is a linearly independent set, then ##\{T(v_1), T(v_2),..., T(v_k) \}## is linearly independent. Homework EquationsThe Attempt at a Solution This seems to be true, because we know that ##a_1v_1 + a_2v_2 + \cdots +...
  20. A

    Question about linear independence

    Homework Statement Homework EquationsThe Attempt at a Solution if there exists a set with 3 vectors, and all of them are linear independent, then by definition no linear combination of the 3 vectors can equal to 0. I believe that's an accurate definition right? So in this case, the answer...
  21. Mr Davis 97

    Linear independence of polynomials of different degree

    Homework Statement Let S be a set of nonzero polynomials. Prove that if no two have the same degree, then S is linearly independent. Homework EquationsThe Attempt at a Solution We will proceed by contraposition. Assume that S is a linearly dependent set. Thus there exists a linear dependence...
  22. Mr Davis 97

    Linear Independence of a Set of Vectors

    Homework Statement Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent. Homework EquationsThe Attempt at a Solution I think that that it would be easier to prove the logically equivalent statement: Prove that a set S of vectors...
  23. Mr Davis 97

    I Linear independence of functions

    Is there a difference between the linear independence of ##\{x,e^x\}## and ##\{ex,e^x\}##? It can be shown that both only have the trivial solution when represented as a linear combination equal to zero. However, the definition of linear independence is: "Two functions are linearly independent...
  24. K

    Math proof: Linear Independence

    Homework Statement How can I show that if a vector (in a vector space V) cannot be written as a linear combination of a linearly independent set of vectors (also in space V) then that vector is linearly independent to the set? Homework Equations To really prove this rigorously it would make...
  25. C

    I Question On Linear Independence

    We were going over linear independents in class and my professor said that if y1 and y2 are linearly independent then the ratio of y2/y1 is not a constant, but he never explained why it is not a constant.
  26. A

    MHB Proving Linear Independence

    In a problem I am working on, it is given that $V_1, V_2, ... , V_n$ are mutually perpendicular vectors in a space defined with a certain scalar product. I need to prove or disprove that $V_i$ are linearly independence regardless of any definition of scalar product. I think the solution should...
  27. Arnab Patra

    Find general equation of x′′(t)+5x′(t)+4x(t)=0

    Suppose ##x_1(t)## and ##x_2(t)## are two linearly independent solutions of the equations: ##x'_1(t) = 3x_1(t) + 2x_2(t)## and ##x'_2(t) = x_1(t) + 2x_2(t)## where ##x'_1(t)\text{ and }x'_2(t)## denote the first derivative of functions ##x_1(t)## and ##x_2(t)## respectively with respect to...
  28. G

    MHB How can the Wronskian be used to determine linear independence?

    I'm asked to check whether $\left\{1, e^{ax}, e^{bx}\right\}$ is linearly independent over $\mathbb{R}$ if $a \ne b$, and compute the dimension of the subspace spanned by it. Google said the easiest way to do this is something called the Wronskian. Is this how you do it? The matrix is: $...
  29. F

    Prove that three functions form a dual basis

    Homework Statement Homework Equations[/B] The Attempt at a Solution From that point, I don't know what to do. How do I prove linear independence if I have no numerical values? Thank you.
  30. kostoglotov

    Need help interpreting the Wronskian

    I'm given bases for a solution space \left \{ x,xe^x,x^2e^x \right \}. Clearly these form a basis (are linearly independent). But, unless I've made a mistake, doing the Wronskian on this yields W(x) = x^3e^x. Isn't this Wronskian equal to zero at x = 0? Isn't that a problem for...
  31. G

    Linear algebra: Find the span of a set

    Homework Statement Find the span of U=\{2,\cos x,\sin x:x\in\mathbb{R}\} (U is the subset of a space of real functions) and V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\} Homework Equations -Vector space span -Linear independence...
  32. Q

    Linear independence of 5 dimensional vectors in R^3

    i am asked to determine whether 3 vectors which have 5 dimensions (x,y,z,w,u) are linearly dependent or independent in R^3. it doesn't make any sense. should i ignore w and u dimensions and take x,y,z only? because if i dont, all answers would be same, doesn't matter in r^3 or R^4 etc. the...
  33. Sollicitans

    Linear Independence of trigonometric functions

    Homework Statement There's no reason to give you the problem from scratch. I just want to show that 5 trigonometric functions are linearly independent to prove what the problem wants. These 5 functions are sin2xcos2x. sin2x, cos2x, sin2x and cos2x. Homework Equations...
  34. W

    Affine independence in terms of linear independence

    This question mostly pertains to how looking at affine independence entirely in terms of linear independence between different families of vectors. I understand there are quite a few questions already online pertaining to the affine/linear independence relationship, but I'm not quite able to...
  35. RJLiberator

    Linear Algebra Proof involving Linear Independence

    Homework Statement Prove that if ({A_1, A_2, ..., A_k}) is a linearly independent subset of M_nxn(F), then (A_1^T,A_2^T,...,A_k^T) is also linearly independent. Homework EquationsThe Attempt at a Solution Have: a_1A_1^T+a_2A_2^T+...+a_kA_k^T=0 implies a_1A_1+a_2A_2+...+a_kA_k=0 So...
  36. R

    Proof: Max number of Linearly Independent Vectors

    Homework Statement Prove that a set of linearly independent vectors in Rn can have maximum n elements. So how would you prove that the maximum number of independent vectors in Rn is n?I can understand why in my head but not sure how to give a mathematical proof. I understand it in terms of the...
  37. nuuskur

    Linear Independence: Answer to Homework

    Homework Statement Assume vectors ##a,b,c\in V_{\mathbb{R}}## to be linearly independent. Determine whether vectors ##a+b , b+c , a+c## are linearly independent. Homework EquationsThe Attempt at a Solution We say the vectors are linearly independent when ##k_1a + k_2b +k_3c = 0## only when...
  38. nuuskur

    Linear Independence of \cos x, \cos (x+2), \sin (x-5)

    Homework Statement Given the system of vectors \cos x, \cos (x+2), \sin (x-5). Determine whether the system is linearly independent. Homework EquationsThe Attempt at a Solution If it were linearly dependent, there would exist a non-trivial linear combination, such that: k_1\cos x + k_2\cos...
  39. B

    Test Tomorrow Linear independence, spanning, basis

    Hello I'm taking linear algebra and have a couple of questions about linear independence, spanning, and basis Let me start of by sharing what I think I understand. -If I have a matrix with several vectors and I reduce it to row echelon form and I get a pivot in every column then I can assume...
  40. B

    MHB Please Critique My Solution Involving Linear Independence, Linear Dependence and Span

    Problem: True or False? If $x$ and $y$ are linearly independent, and if $\{\textbf{x}, \textbf{y}, \textbf{z}\}$ is linearly dependent, then $\textbf{z}$ is in Span $\{\textbf{x},\textbf{y}\}$ Solution: $\textbf{True}$. If $a\textbf{x} + b\textbf{y} = \textbf{0}$ is true and if $a\textbf{x} +...
  41. A

    Are the Functions Linearly Independent Based on the Matrix of Their Outputs?

    There's a question in charles curtis linear algebra book which states: Let ##f1, f2, f3## be functions in ##\mathscr{F}(R)##. a. For a set of real numbers ##x_{1},x_{2},x_{3}##, let ##(f_{i}(x_{j}))## be the ##3-by-3## matrix whose (i,j) entry is ##(f_{i}(x_{j}))##, for ##1\leq i,j \leq 3##...
  42. N

    Linear Independence Question

    Homework Statement Determine all values of the constant k for which the given set of vectors is linearly independent in \mathbb R^4. {(1, 1, 0, −1), (1, k, 1, 1), (4, 1, k, 1), (−1, 1, 1, k)} Homework Equations The Attempt at a Solution So far I set up a coefficient matrix...
  43. kq6up

    Wronskian vs. Determinant in Determining Linear Independence?

    It seems to me that if a row is able to be zeroed out through Gaussian reduction that the determinate of that matrix would equal zero. Therefore, we know that at least one of equations/vectors that constructed the matrix was formed from the other two rows. That is -- that equation is dependent...
  44. N

    Checking for linear independence of certain vectors

    Homework Statement Given that { u1, u2, u3, u4, u5, u6 } are linearly independent vectors in R16, and that w is a vector in R16 such that w ∉ span{ u1, u2, u3, u4, u5, u6 }. a) Is the set { 0, u1, u5 } is linearly independent? b) the set { u1, u2, u3, u4, u5, u6, w } is linearly...
  45. Z

    Proving Linear Independence of vectors

    Homework Statement Let x1 = (1, 2, -1, 1), x2 = (-1, -1, -1, -1), x3 = (1, 1, 1, 0), x4 = (-2, -1 -4 -1) Show that x1, x3 and x4 are linearly independent Homework Equations The Attempt at a Solution Now I used the equation: ax1+bx2+cx3+dx4=0 Hence forth the augmented...
  46. stripes

    Linear Independence: Homework Equations & Solutions

    Homework Statement Homework Equations The Attempt at a Solution For part (a): a*1 + b*√2 + c*√3 = 0 assume a, b, c not all zero a + b√2 = -c√3 a2 + 2b2 + 2ab√2 = 3c2 a2 + 2b2 - 3c2 = -2ab√2 (a2 + 2b2 - 3c2)/(-2ab) = √2 which is not possible since we take a, b, c to...
  47. Petrus

    MHB Linear Independence of Vectors: Why Determinant ≠ 0?

    Hello MHB, I got one question. If we got this vector V=(3,a,1),U=(a,3,2) and W=(4,a,2) why is it linear independence if determinant is not equal to zero? (I am not interested to solve the problem, I just want to know why it is) Regards, |\pi\rangle
  48. Fernando Revilla

    MHB Tyler's question at Yahoo Answers (linear independence)

    Here is the question: Here is a link to the question: Linear algebra help please please? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  49. Fernando Revilla

    MHB RF's question at Yahoo Answers (linear independence, Wronskian).

    Here is the question: Here is a link to the question: Differential Equations...Linear independence question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  50. mef51

    Checking Linear Independence. Using Wronskian vs. Using Definition

    Homework Statement Is the set $$ \{cos(x), cos(2x)\} $$ linearly independent?Homework Equations Definition: Linear Independence A set of functions is linearly dependent on a ≤ x ≤ b if there exists constants not all zero such that a linear combination of the functions in the set are equal to...
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