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. B

    Linear Independence. express each vector as a lin. combo

    V1 = (1,2,3,4) V2 = (0,1,0,-1) V3 = (1,3,3,3) a) I already expressed them a linearly dependent set in R4 b) Express each vector in part (a) as a linear combination of the other two linear combo is just {c1v1 + c2v2...cnvn} right? But I don't get where to start to prove this
  2. S

    Proving Linear Independence of 2nd DE Solutions

    Homework Statement suppose v(t) , u(t) are two linearly independent solution of the 2nd DE. (6t^2-t-1)y''+t^2e^ty'-(3t^3-t-1)y=t^2e^t-3t^3+1 satisfying the condition v(0)=u(0)=1 , prove that u'(0) ≠ v'(0) Homework Equations The Attempt at a Solution I've tried to use Wronskian...
  3. B

    Linear Independence: Is this true?

    Consider a plane P in ℝ^{3}. Is it necessarily the case that any vector outside this plane cannot be expressed as a linear combination of finitely many vectors on this plane? I would think yes; if you tried to parametrize the plane P with two parameters, could we somehow show that there are...
  4. B

    Linear Independence of vectors question

    Homework Statement Suppose that A, B and C are not linearly independent. Then show how the a_i can be computed, up to a common factor, from the scalar products of these vectors with each other Homework Equations a_1A + a_2B + a_3C = 0 a_1=a_2=a_3=0 Hint - Suppose that there are non-zero...
  5. M

    Linear Algebra: Span, Linear Independence Proof

    Homework Statement Suppose v_1,v_2,v_3,...v_n are vectors such that v_1 does not equal the zero vector and v_2 not in span{v_1}, v_3 not in span{v_1,v_2}, v_n not in span{v_1,v_2,...v_(n-1)} show that v_1,v_2,v_3,...,V_n are linearly independent. Homework Equations linear independence...
  6. P

    Span, linear independence problem

    The problem is attached. I don't know why he called all 4 vectors V1, I guess it was a typo. Anyways, part I) This is not linearly independent as the determinant of the matrix containing those 4 vectors is 0 I am having trouble with part II) I think I know the answer, but I don't...
  7. D

    Wronskian and linear independence

    Hello, I understand that if we have three functions f, g, and h, they are linearly independent <=> the only c1, c2, and c3 that satisfy (c1)f+(c2)g+(c3)h=0 are c1=c2=c3=0. In order to solve for these c1, c2, and c3, we want three equations in the three unknowns. To do this we can...
  8. caffeinemachine

    MHB Linear algebra. Rank. linear independence.

    Let $V$ be a finite dimensional vector space. Let $T$ be a linear transformation on $V$ with eigenvalue $0$. A vector $v \in V$ is said to have rank $r > 0$ w.r.t eigenvalue $0$ if $T^rv=0$ but $T^{r-1}v\neq 0$. Let $x,y \in V$ be linearly independent and have ranks $r_1$ and $r_2$ w.r.t...
  9. A

    Discussion problem, wronskian matrix, linear independence of solution

    Edit: I think I may have posted this in the wrong section, sorry about that. Note that this isn't a homework problem though, I"m not enrolled in this class, I was just reading over some of this stuff and trying some problems since I"m majoring in physics. I have a textbook "discussion" problem...
  10. srfriggen

    Linear Independence of functions

    Homework Statement This is from Serge Lang's "Linear Algebra, 3rd Edition", page 15. Consider the vector space of all functions of a variable t. Show that the following pairs of functions are linearly independent: (a) 1,t (b) t, t2 (c) t, 4 Homework Equations...
  11. srfriggen

    Linear Independence of two functions and differentiation

    This is from my text, "Linear Algebra" by Serge Lang, pg 11: -The two functions et, e2t are linearly independent. To prove this, suppose that there are numbers a, b such that: aet + be2t=0 (for all values of t). Differentiate this relation. We obtain aet + 2be2t = 0. Subtract...
  12. S

    Verifying whether my working is correct in showing Linear Independence

    Homework Statement I have attempted the questions below but am not sure if I am applying the method correctly to show linear dependence/independence. a)Show that the vectors e1=[1 1 0]T, e2=[1 0 1]T, e3=[0 1 1]T are linearly independent b) Show that the vectors e1=[1 1 0]T, e2=[1 0 -1]T...
  13. L

    Linear independence of sin (x), cos (x) and 1, proof

    What would be the best way to show that functions f(x)=1, g(x)=sin(x) and h(x)=cos(x) are linearly independent elements of the vector space \mathbb{R}^{\mathbb{R}}? I know that the linear independence means that an expression like \alpha \mathbb{x}_1 + \beta \mathbb{x}_2 + \gamma \mathbb{x}_3...
  14. R

    Another method? - Matrix & Linear Independence

    Question : Let A be a 7 × 4 matrix. Show that the set of rows of A is linearly dependent. Answer: The row vectors of a matrix are linearly independent if and only if the rank of the matrix is equal to the number of rows in the matrix. Since rank (A) = 4 , and the number of rows in the...
  15. D

    Linear Algebra: Linear Independence

    Homework Statement Let S be a basis for an n-dimensional vector space V. Show that if v1,v2,...,vr form a linearly independent set of the vectors in V, then the coordinate vectors (v1)s, (v2)s,...,(vr)s form a linearly independent set in the Rn, and conversely. Homework Equations...
  16. B

    MHB Linear independence of polynomial set.

    Hi guys, I've been working on a question which is as follows: For which real values of c will the set $\{1+cx, 1+cx^2, x-x^2\}$ be a basis for $P_2$? I'm coming up with the answer as no values of c, but am I really wrong? I've only checked linear independence, because it would imply that it...
  17. 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...
  18. J

    Determining linear independence

    1. Homework Statement If set A={u,v,w} ⊂ R^n is linearly independent, is B={u-v, u+w, v+w}⊂ R^n linearly independent? 2. Homework Equations 3. The Attempt at a Solution Since A is linearly independent, there exist no all non-zero scalars a1, a2, a3 such that a1*u+a2*v+a3*w=0...
  19. S

    Linear Independence: What It Is & When to Use

    Any one please tell me about the term linear independence?and when we say that the function is linear independent
  20. S

    How do I test for linear independence if there are only 3 equations in R4?

    Homework Statement So the dimension is R4. V1=[3 1 1 2], V2=[-2 -1 2 2] and V3=[2 1 2 1] Homework Equations The Attempt at a Solution The only way I know of to test for convergence is to make a matrix out of the row vectors of the vectors above (with the row vectors becoming the...
  21. E

    Linear algebra help: Linear independence

    Homework Statement Let A be an m x n matrix of rank n. Suppose v_1, v_2, ..., v_k \in \mathbb{R}^n and \{v_1, v_2, ..., v_k\} is linearly independent. Prove that \{Av_1, Av_2, ..., Av_k\} is likewise linearly independent. Homework Equations The Attempt at a Solution It says I...
  22. S

    Test the set of functions for linear independence in F

    Homework Statement Test the set of {1, ln(2x), ln(x^2)} for linear independence in F, the set of all functions. If it is linearly dependent, express one of the functions as a linear combination of the others. Homework Equations N/A The Attempt at a Solution I know if [ a(1)...
  23. T

    Linear independence if there is a column of zeros

    Homework Statement If, in a matrix, there is a column of all zeros, does this mean the given vector/matrix is linearly dependent? An example would be: [1 2 0 4] [2 3 0 1] [5 2 0 7] A few questions to clear up some possible misconceptions: 1) The matrix above is a 4-dimensional vector...
  24. S

    Linear Independence of \overline{w} and \overline{v} in R4/U

    Homework Statement Well it isn't so much the problem as it is the notation used within the problem. But here is the question: Determine whether or not \overline{w} and \overline{v} are linearly independent in R4/U Homework Equations If v \in V then \overline{v} = v + U The...
  25. D

    Linear independence in normed vector space

    Homework Statement If we have a normed vector space, and a sequence of vectors \{\mathbf{v}_k\}_{k=1}^{N} in the normed vector space. If there exists a constant B>0 such that the following holds for all scalar coefficients c_1,c_2\cdots c_N B\sum\limits_{k=1}^N |c_k|^2 \leq...
  26. T

    Linear Independence and Intersections of Sets

    Homework Statement Let E' and E'' be linearly independent sets of vectors in V. Show that E' \cap E'' is linearly independent. The Attempt at a SolutionTo show a contradiction, let E' \cap E'' be linearly dependent. Also let A be all of the vectors in E' \cap E''. Thus, A \subseteq E' and A...
  27. C

    Conditions for linear independence

    Homework Statement Under what conditions on the numbers a and b are the vectors (1,a), (1,b) linearly independent in R2? Homework Equations The Attempt at a Solution x(1,a)+y(1,b)=(0,0) (x,ax)+(y,by)=(0,0) (x+y, ax+by)=(0,0) x+y=0, ax+by=0 x=-y, ax=-by unsure where to go from...
  28. H

    [Linear Algebra] Linear Independence

    Homework Statement Show if S = {v1,v2,v3} is independent or dependent . . . Homework Equations (0,0,0,0) = k1(a,b,c,d) + k2(e,f,g,h) + k3(i,j,k,l) where {a,b,c,d,e,f,g,h,i,j,k,l \in ℝ} The Attempt at a Solution im trying to tell if i can say that this set of 3 vectors in R4 is...
  29. A

    What makes the set {1,x,x^2,...,x^n,...} linearly independent in Q[x]?

    Homework Statement Show that the set {1,x,x^2,...,x^n,...} is linearly independent in Q[x]. The Attempt at a Solution Since an infinite set of vectors is linearly independent if each finite subset is also linearly independent, I think I need to show that every subset of...
  30. A

    Does differentiation preserve linear independence?

    If we take the derivative of n functions that are linearly independent to each other and we write it down like c1f1(x) + c2f2(x) +...+ cnfn(x)=0, then would the linear independence be preserved if we differentiate the equation with respect to x?
  31. J

    Genericity of Linear Independence

    Restrict attention to vectors in ℝ^m where m is a natural number. Let σ be the vector of ones. Let V be the set of vectors whose largest entry is 1 and whose smallest entry is 0. When is it (generically) the case that the set of vectors {σ, v_1, v_2, ..., v_n} is linearly independent...
  32. M

    Linear independence and decompostion

    Explain why the method of decomposition when applied to the solution set of a homogeneous linear system always yields a linearly independent set of vectors whose span is the set of solutions... Can someone explain this it seems reasonable but I can't seem to prove it to myself
  33. S

    Linear Independence: Showing 1, sin^2(x), sin(2x) is Independent

    Homework Statement there is the vector space F(R) = {f | f:R -> R } show that {1, sin^2(x), sin(2x)} is linearly independent Homework Equations a(1) + b(sin^2(x)) + c(sin(2x)) = 0, where the ONLY solution is a=b=c=0, for the set to be implied linearly independent.The Attempt at a Solution...
  34. P

    Wronskian to prove linear independence

    Homework Statement Let v_1,v_2 be any two solutions of the differential equation y''+ay'+by=0 such that \frac {v_2}{v_1} is not constant, and let f(x) be any solution of the differential equation as well. Use the properties of the Wronskian to prove that constants c_1,c_2 exist such that: c_1...
  35. M

    LA - Proof Involving Linear Independence and Spanning

    Homework Statement I was trying to prove a theorem from Axler's Linear Algebra text and my proof is different from the one in the book, and I'm wondering if someone can check whether or not my proof works, since I'm just starting to write proofs. Theorem 2.6 (pg. 25): In a finite-dimensional...
  36. T

    Linear Independence of subsets

    Homework Statement Suppose {V1, V2, ..., Vp} form a linearly independent set of vectors. Show that any subset of this collection of vectors is also linearly independent. Is it necessarily true that is the vectors are dependent, that any subset is also dependent? Homework Equations The...
  37. T

    Linear Independence, Differential Equations

    Homework Statement [PLAIN]http://img220.imageshack.us/img220/7427/diff5.jpg The Attempt at a Solution Done (a). How do I go about (b) and (c)?
  38. AlexChandler

    Is Linear Independence Preserved Under Subsets?

    Homework Statement Let V be a vector space and \{v_1,...,v_{n+1} \} \subset V a set of linearly independent vectors of V . Show directly: (Don't just quote a theorem!) (a) The set \{v_1,...,v_{n} \} is linearly independent. (b) v_{n+1} \not \in span \{v_1,...,v_{n} \} Homework...
  39. J

    Is This a Correct Approach to Determining Linear Independence?

    Homework Statement Critique my understanding. Homework Equations From the omniscient Wikipedia: The Attempt at a Solution So if I had (0 0 1)T, (0 2 -2)T, (1 -2 1)T, and (4 2 3)T, then I'd check whether at least one of them can be written as a linear combination of the others by looking...
  40. K

    Linear Independence Proof

    I'm trying to finish these linear independence proofs: 3. Let S = {v1, v2, v3} be a linearly independent subset of V and let T = {v1 + v2, v2 + v3, v1 + v3}. (a) Show that if char F is not 2, then T is linearly independent. (b) Show that if char F = 2, then T is not linearly independent. 4...
  41. H

    Linear Independence of t, e^t, sin(t) in Vector Space V

    Let V be the vector space of all real-valued continuous functions. t, e^t, sin(t) are in V. Is t, e^t, sin(t) in V linearly independent? My answer is yes. However, how can I prove it which is that which do I have to show or can I just say the def of linear independent?
  42. H

    Linear Independence: Proving Dependence & Independence of Vectors, Sets

    1) Let u and v be nonzero vectors in a vector space V. show that u and v are linearly dependent if and only if there is a scalar k such that v = ku. Equivalently, u and v are linearly independent if and only if neither vector is a multiple of the other. 2) Let S = {v1, v2, ..., vk} be a set of...
  43. M

    Linear algebra: subspaces, linear independence, dimension

    Homework Statement 1. Consider three linearly independent vectors v1, v2, v3 in Rn. Are the vectors v1, v1+v2, v1+v2+v3 linearly independent as well? 2. Consider a subspace V of Rn. Is the orthogonal complement of V a subspace of Rn as well? 3. Consider the line L spanned by [1 2...
  44. A

    Proof of linear independence and dependence

    1. Homework Statement There are two proofs: Let X and Y be two matrices such that the product XY is defined. Show that if the columns of Y are linearly dependent, then so are the columns of the matrix XY. Let X and Y be two matrices such that the product XY is defined. Show that if...
  45. P

    Linear Independence of Polynomials

    Homework Statement Given a set of polynomials in x: x^{r_1}, x^{r_2},...,x^{r_n} where r_i \neq r_j for all i \neq j (in other words, the powers are distinct), where the functions are defined on an interval (a,b) where 0 < a < x < b (specifically, x \neq 0), I'd like to show that this...
  46. Saladsamurai

    Linear Independence: Is the set LI?

    Homework Statement Give that u and v are LI and that u and w are LI and that v and w are LI, is the set {u,v,w} LI ? Prove or disprove. The Attempt at a Solution I know that this can be done by providing a counterexample. But I wanted to know if there is a way to prove it generally? That...
  47. K

    Linear independence with differentiable functions

    I don't this this is an overly complicated proof but it is one I have never seen or done before. Let f be a polynomial with atleast two non-zero terms having different degrees. Prove that the set {f(x),xf'(x)} is linearly independent in P Proof: With out loss of generality we can...
  48. L

    Do 4 Linearly Independent Vectors in R^4 Always Span the Space?

    Homework Statement You are given 4 vectors in R^4 which are linearly independent. Do they always span R^4?Homework Equations The Attempt at a Solution Intuitively, I think the answer is yes. I know if I want to show they span R^4, I need to use the general terms, but all I can think of is the...
  49. Z

    Can Linear Independence be Proven with Given Information?

    Homework Statement [PLAIN]http://uploadpie.com/nsXSv Homework Equations The Attempt at a Solution I have no idea how to start. To be linearly independent, c1u1+c2u2+...+cnun = 0 has only trivial solution. But I don't know how can I use the given information to prove that
  50. T

    Show W Belongs to Span of {X, Y, Z}: Linear Independence Homework

    Homework Statement Let {X, Y, Z} be linearly independent in Rn. If {X, Y, Z, W} is linearly dependent, show that W \epsilon span{X, Y, Z}. NB: You must SHOW this. Homework Equations The Attempt at a Solution For W to belong to the span of {X,Y,Z}, W = aX + bY + cZ where a, b, c...
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