# What is Linear dependence: Definition and 67 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. ### I Are Maxwell's equations linearly dependent?

HI, consider the 4 Maxwell's equations in microscopic/vacuum formulation as for example described here Maxwell's equations (in the following one assumes charge density ##\rho## and current density ##J## as assigned -- i.e. they are not unknowns but are given as functions of space and time...
2. ### Determining value of r that makes the matrix linearly dependent

for problem (a), all real numbers of value r will make the system linearly independent, as the system contains more vectors than entry simply by insepection. As for problem (b), no value of r can make the system linearly dependent by insepection. I tried reducing the matrix into reduced echelon...
3. ### 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}...
4. ### Matrix concept Questions (invertibility, det, linear dependence, span)

I have a trouble showing proofs for matrix problems. I would like to know how A is invertible -> det(A) not 0 -> A is linearly independent -> Column of A spans the matrix holds for square matrix A. It would be great if you can show how one leads to another with examples! :) Thanks for helping...
5. ### MHB Are These Vectors Linearly Dependent?

Are the vectors $$\left[ \begin{array}{r} 2\\1\\-2 \end{array}\right] ,\quad \left[\begin{array}{r} 0\\2\\-2 \end{array}\right] ,\quad \left[\begin{array}{r} 2\\3\\-4 \end{array}\right]$$ linearly dependent or linearly independent...
6. ### I Function of 2 variables, max/min test, D=0 and linear dependence

##f(x,y)## a critical point is given by ##f_x=0## and ##f_y=0## simultaneously. the test is: ##D=f_{xx}f_{yy}-(f_{xy})^2 ## if ##D >0 ## and ##f_{xx} <0 ## it is a max if ##D >0 ## and ##f_{xx} >0 ## it is a min ##D >0 ## is is a saddle if ##D =0 ## it is inconclusive, and ##f_x## and...
7. ### I Regarding the linear dependence of eigenvectors

Let's say we have a set of eigenvectors of a certain n-square matrix. I understand why the vectors are linearly independent if each vector belongs to a distinct eigenvalue. However the set is comprised of subsets of vectors, where the vectors of each subset belong to the same eigenvalue. For...
8. ### Linear Dependence and Non-Zero Coefficients

Homework Statement True or False: If u, v, and w are linearly dependent, then au+bv+cw=0 implies at least one of the coefficients a, b, c is not zero Homework Equations Definition of Linear Dependence: Vectors are linearly dependent if they are not linearly independent; that is there is an...
9. ### How to correctly solve this problem? (linear dependency)

This is the problem: Suppose a, b and c are linearly independent vectors. Determine whether or not the vectors (a + b), (a - b), and (a - 2b + c) are linearly independent. Here was my solution, which involved writing words (and hasn't actually been confirmed correct yet): Let's align a, b and...
10. ### I Same vector space for arbitrary independent vectors?

If we use n linearly independent vectors x1,x2...xn to form a vector space V and use another set of n linearly independent vectors y1,y2...yn to form a vector space S, is it necessary that V and S are the same? Why? If we have a vector space Q that the dimension is n, can we say that any set of...
11. ### Linear dependence of functions

Homework Statement check for linear dependecy[/B] f(x) = cosx and g(x) = xcosx 2 functions from R to R Homework EquationsThe Attempt at a Solution Why this is wrong: if i take the scalar a1 = 3, a2 = 1 i can do that since 3 is real, and a1 is in R. so 3f(3) + -1g(3) = 0 there for we have none...
12. ### I Linear dependence of functions

Functions f,g from R to R. f(x) = xcosx, g(x) = cosx for x = 0, we get f(x) = 0, g(x) = 1. so for scalar t in R t(f(x)) + 0 * g(x) = 0 . ==> f(x) and g(x) are linearly idepenent. Is that right? if so in functions we search for an x that makes the function dependent?
13. ### MHB Vector Spaces .... Linear Dependence and Indepence .... Basic Proof Required

In Andrew McInerney's book: First Steps in Differential Geometry, Theorem 2.4.3 reads as follows:https://www.physicsforums.com/attachments/5252McInerney leaves the proofs for the Theorem to the reader ... I am having trouble formulating a proof for Part (3) of the theorem ... Can someone help...
14. ### Proving dependent columns when the rows are dependent

I feel like I almost understand the solution I've come up with, but a step in the logic is missing. I'll post the question and my solution in LaTeX form. Paraphrasing of text question below in LaTeX. Text question can be seen in its entirety via this imgur link: http://i.imgur.com/41fvDRN.jpg...
15. ### Linear dependence of two vectors

Suppose the vectors ##v_a## and ##v_b## are linearly independent, another vector ##v_c## is linearly dependent to both ##v_a## and ##v_b##. Now if I form a new vector ##v_d##, where ##v_d = v_b+cv_c## with ##c## a constant, will ##v_d## be linearly independent to ##v_a##? I need to check how I...
16. ### System of vectors, linear dependence

Homework Statement Prove that if in a system of vectors: S_a =\{a_1, a_2, ..., a_n\} every vector a_i is a linear combination of a system of vectors: S_b = \{b_1, b_2, ..., b_m\}, then \mathrm{span}(S_a)\subseteq \mathrm{span}(S_b) Homework EquationsThe Attempt at a Solution We know due to...
17. ### Proof that det(M)=0 => Linear Dependence of Columns

Homework Statement Prove that for a general NXN matrix, M, det(M)=0 => Linear Dependence of Columns Homework EquationsThe Attempt at a Solution It's not clear to me at all how to approach this. We've just started Linear algebra and this was stated without proof in lecture. I have no idea how...
18. ### 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} +...
19. ### What does it mean for a set of vectors to be linearly dependent?

Hi all, I was asked by someone today to explain the notion of linear independence of a set of vectors and I would just like to check that I explained it correctly. A set of vectors S is said to be linearly dependent if there exists distinct vectors \mathbf{v}_{1}, \ldots , \mathbf{v}_{m}...
20. ### Linear Dependence in Rn with Nonsingular Matrix A

Homework Statement Let x1,x2,x3 be linearly dependent vectors in Rn, let A be a nonsingular n x n matrix, and let y1=Ax1, y2=Ax2, y3=Ax3. Prove that y1, y2,y3 are linearly dependent. Homework Equations The Attempt at a Solution My solution was y is equal to the zero vector...
21. ### MHB Proving Linear Dependence of r1,r2,r3 Given a,b,c ≠ 0

Given that r1=2a-3b+c r2=3a-5b+2c r3=4a-5b+c where a, b, c are non-zero and non coplannar vectors How to prove that r1, r2 , r3 are linearly dependent? I have moved with c1*r1+c2*r2+c3*r3=0 but confused how to show that at leat one of c1, c2, c3 is non-zero. We only have the information...
22. ### Linear Dependence and Span Question

Homework Statement Is the following set linearly dependent or independent? And does this set span the given space? {eX, e-x}\inC∞(R) Homework Equations The Attempt at a Solution So, if it's linearly independent, then: k1ex +k2e-x = 0 where k1,k2=0 and only 0. But if you let k1=...
23. ### MHB Linear dependence of polynomical functions

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
24. ### Linear Dependence of Vectors in R^3 with Variable Coefficients

Homework Statement For which real values of \lambda do the following vectors form a linearly dependent set in \mathbb{R}^{3} v_{1}=(\lambda ,-\frac{1}{2},-\frac{1}{2}), v_{2}=(-\frac{1}{2},\lambda ,-\frac{1}{2}), v_{3}=(-\frac{1}{2},-\frac{1}{2},\lambda )The Attempt at a Solution I know that...
25. ### Are These Vectors Linearly Dependent?

Homework Statement http://imgur.com/P9udvTs Homework Equations The Attempt at a Solution So I set the scalar multiples of a, b, and c as x,y,z so i had 3 equations 4x-4y+4z=0 -4x+4y-4z=0 -2x-4y-5z=0 i tried solving it numerous times each time trying to use a different...
26. ### MHB Linear Dependence of Vectors Spanning a Space: Example Needed

Hello A base of some space is a set of vectors which span the space, and are also linearly independent. I am looking for an example of vectors which DO span some space, but are dependent and thus not a base...can anyone give me a simple example of such a case ? thanks !
27. ### What are the components of u-w?

Let u=[1 2 3]T , v=[2 -3 1]T , and w=[3 2 -1]T. Find the components of a) u-w b) 7v+3 c) -w+v d) 3(u-7v) e) -3v-8w f) 2v-(u+w)
28. ### Linear dependence of non-numerical objects

Homework Statement Suppose V = 2^Ω where Ω = {red,blue,yellow,green}. Verify whether u={blue,green}, v={red,yellow,green}, and w={blue,yellow} are linearly independent in V The Attempt at a Solution I let red = a1 blue = a2 yellow =a3 green = a4 therefore Ω = {a1, a2, a3, a4}...
29. ### Find linear dependence on these vectors

Homework Statement Suppose V = R^4 and let U = <X>, where X = {(1,0,-2,1),(2,-2,0,3),(0,2,-4,-1),(-1,2,-2,-2)} Find linear dependence on X and use it to find a smaller generating set of U. Repeat the step until you reach a basis for U. Homework Equations The Attempt at a Solution...
30. ### Solving Linear Dependence in a Vector Space

Hi. I attached the problem and my work. I'm not sure if I did part a) right. In the past problems I've done, they usually provide you with 3 vectors that are linearly independent, thus giving you unique values for C1, C2, C3. The matrix for this one forms: 1 1 1 0 1 3 0 0 0 Which is...
31. 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?
32. ### Linear Dependence in High-Dimensional Vector Spaces

Homework Statement Let S = {v_{1}, v_{2}, \cdots , v_{n}} S is linear dependent iff at least one v in S is a linear combination of the others. Homework Equations The Attempt at a Solution From here on, just take v to be a vector, and x to be some scalar please. I really just wanted to check...
33. ### Row redcing a matrix to determine linear dependence?

Homework Statement Determine if vectors <1,-1,0,2,3> <1,0,-1,3,3> <1,-1,0,3,0> <0,1,-1,2,-2> are linearly dependent or independentHomework Equations I have been solving these questions in the book using a matrix and row reducing them. If I wound up with a free variable, I determined the...
34. ### Linear Dependence of f and g on 1<x<∞

Homework Statement Determine if the pair of functions given are linearly independent or linearly dependent on the interval 1<x<∞, and give a reason for your answer. y1=|x| y2=-3x Homework Equations I'm pretty sure this has something to do with the Wronskian. W(f,g)=fg'-f'g The...
35. ### Problem with linear dependence: det(a)=0, but rref is inconsistent?

Homework Statement v1=[2 1 1 4 2] v2=[-1 2 2 1 -1] v3=[3 -2 1 -2 2] v4=[4 1 4 3 3] v5=[1 2 3 2 1] Find if the system is linear dependent or independent. If it is dependent, express the last vector in the list (v5) as a combination of the preceding ones. Homework Equations...
36. ### Linear Dependence of Vectors: a, b, c

Homework Statement Are the vectors a = [1 -1 0 1], b = [1 0 0 1] and c = [0 -1 0 1] linearly independent? The Attempt at a Solution I am mainly confused about whether or not I should have my matrix in row or column form to solve this: r 1 -1 0 1 s 1 0 0 1 t 0 -1 0 1 or r...
37. ### Linear Dependence and Subsets: Proving Linear Dependence in Sets of Vectors

Homework Statement Suppose that E,F are sets of vectors in V with E \subseteq F. Prove that if E is linearly dependent, then so is F. The Attempt at a SolutionRead post #2. This proof, I think, was incorrect. If we suppose that E is linearly dependent, then we know that there exists...
38. ### Proving linear dependence (2.0)

I have to prove: Consider V=F^{n}. Let \mathbf{v}\in V/\{e_{1},e_{2},...,e_{n}\}. Prove \{e_{1},e_{2},...,e_{n},\mathbf{v}\} is a linearly dependent set. My attempts at a proof: Since {e_{1},e_{2},...,e_{n}} is a basis, it is a linearly independent spanning set. Therefore, any vector...
39. ### How Can We Prove Linear Dependence in Vector Spaces?

I have to prove: Let u_{1} and u_{2} be nonzero vectors in vector space U. Show that {u_{1},u_{2}} is linearly dependent iff u_{1} is a scalar multiple of u_{2} or vice-versa. My attempt at a proof: (\rightarrow) Let {u_{1},u_{2}} be linearly dependent. Then, \alpha_{1}u_{1}+...
40. ### Linear dependence of a set under linear transformation?

Hi all, Here is the problem: If T: V -> W is a linear transformation and S is a linearly dependent subset of V, then prove that T(S) is linearly dependent. Now, I know that the usual proof goes as follows: Since S is linearly dependent, there are distinct vectors v_1, ..., v_n in S and...
41. ### Check Linear Dependence/Independence of Vectors Without Calculator

Does anyone know a good way to check if a given set of vectors (assume we just know we have a set, not their values) is linearly dependent or linearly independent without a calculator? Ex: Given a set of n-dimensional vectors, say, vector1, vector2, and vector3, how would one determine if these...
42. ### Linear Dependence of Matrix Vectors

I need to prove that the set {I, A, A^2,..., A^n} is linear dependent where A is any nxn matrix. The vector space is the set of nxn matrix, considered as a nxn dimensional vector space. Does anybody have an idea how to prove it? Thank you very much.
43. ### Show linear dependence directly

Homework Statement Show directly that the given functions are linearly dependent on the real line==>find a non-trivial linear combination of the given functions that vanishes identically. Homework Equations f(x) = 2x, g(x) = 3x^{2}, h(x) = 5x-8x^{2} The Attempt at a Solution I...
44. ### Linear dependence of square matrices

I am studying the subject of linear dependence right now and had a question on this topic. Is it possible to construct a square matrix A such that the columns of A are linearly dependent, but the columns of the transpose of A are linearly independent? My intuition tells me no, but I'm not sure...

Do all linearly dependent sets have elements that are linear combinations of each other? Or does this apply only to some of the Linearly Dependent sets? And as a follow up question: How do you know if a set of 2x2 matrices is linearly dependent or linearly independent? Thank you and may...
46. ### Linear Transformation and Linear dependence - Proof

Homework Statement Let T:Rn to Rm be a linear transformation that maps two linearly independents vectors {u,v} into a linearly dependent set {t(u),T(v)}. Show that the equation T(x)=0 has a nontrivial solution. Homework Equations c1u1 + c2v2 = 0 where c1,c2 = 0 T(c1u1 + c2v2) = T(0)...
47. ### Unraveling Ince's Wronskian Puzzle: Linear Dependence of Minors

I'm reading Ince on ODEs, and I'm in the section (in Chapter 5) where he talks about the Wronskian. There are quite a few things here that I don't quite understand or follow. I'm not going to get into all the details, but briefly, suppose we have the Wronskian of k functions: W =...
48. ### Linear Dependence Proof for a Set of Vectors

given S is a set of vectors S= (v1,v2,..vn), prove that S is linearly dependent if and only if one of the vectors in S is a linear combination of all the other vectors in S? Can someone point me in the right direction of how to start this proof? I am completely lost.
49. ### Linear Dependence of x1, x2 and x3 in R^2

x1= column vector (2, 1) x2= column vector (4, 3) x3= column vector (7, -3) Why must x1, x2, and x3 be linearly dependent? x1 and x2 span R^2. The basis are these two columns vectors: (3/2, -1/2), (-2, 1) Since x1 and x2 form the basis, x3 can be written as a linear combination of...
50. ### Linear dependence and Wronskian

Homework Statement Okay so the question is to show that these 2 functions are linearly dependent. ie. they are not both solutions to the same 2nd order, linear, homogeneous differential equation for non zero choices of, say M, B and V Homework Equations f(x) = sin(Mx) g(x) = Bx + V...