# 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. ### 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. ### 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. ### 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. ### 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. ### 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}...
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### 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. ### 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. ### 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. ### 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...
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### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### MHB Are Mutually Perpendicular Vectors Always Linearly Independent?

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. ### 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...

41. ### 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##...
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### What Values of k Ensure Linear Independence in R^4 for These Vectors?

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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...