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. C

    Linear independence after change of basis

    Will a set of vectors stay linearly independent after a change of basis? If it's not always true then is it likely or would you need a really contrived situation?
  2. T

    Linear independence of columns of a matrix

    Homework Statement Are the columns of this matrix linearly independent? 1...3...-2 0...-8...11 0...0...1 0... 0... 0 (periods are just to make spacing clear) The Attempt at a Solution What is confusing me is the last row of zeros. If a set of vectors contains the zero vector, it is linearly...
  3. Jack Nagel

    Linear Independence and Spanning

    Say that {W1, W2, W3, W4} is linearly independent in R4. Now say I have this vector [ 2 tan(h) 7 4sec(k) ] and I want to find values of h and k such that it is not in the span of (W1...W4). If I understand this correctly, it means it is impossible to find those values since they do not...
  4. Y

    Linear independence and dependence

    Hi everyone, having problems with this question, can anyone please help Question: consider a 2 x 2 matrix, can you construct a matrix whose columns are linearly dependent and whose rows are linearly independent? My answer is no. I cannot think of any combination that would make this true...
  5. Q

    Spanning sets, and linear independence of them

    I've become sort of confused on the topic of the linear span versus spanning sets. I know that the span of a subset is the set containing all linear combinations of vectors in V. Is a spanning set then the same thing, or is it something else? Also, in terms of bases... A basis is a linearly...
  6. radou

    Linear Independence of Subsets: Necessary and Sufficient Conditions

    Let V be a vector space over a field F, v_{1}, \cdots, v_{n} \in V and \alpha_{1}, \cdots, \alpha_{n} \in F. Further on, let the set \left\{v_{1}, \cdots, v_{n}\right\} be linearly independent, and b be a vector defined with b=\sum_{i=1}^n \alpha_{i}v_{i}. One has to find necessary and...
  7. radou

    Proving Linear Independence in Vector Spaces with Ordered Sets

    I need to check the proof of the proposition below we got for homework, thanks in advance! Proposition. Let V be a vector space over a field F, and S = \left\{a_{1}, \cdots, a_{k}\right\}\subset V, k\geq 2. If the set S is linearly dependent, and a_{1} \neq 0, and if we assume there is an...
  8. A

    Proving Linear Independence in a Subset of Trigonometric Functions

    I'm stuck on a question in linear algebra, it reads "Show that the subset S={cos mx, sin nx: m between 0 and infinity, n between 1 and infinity} is linearly independent. I really just don't know where to start. I've seen a similar question which was just sin (nx) and the lecturer integrated...
  9. V

    Linear Independence of 3 Vectors in R^4

    LINEAR ALGEBRA: 3 vecotrs in R^4 (with 6 variables) -- Are they linearly independent? For which values of the constants a, b, c, d, e, anf f are the following vectors linearly independent? Justify your answer...
  10. S

    Complicated definitions of linear independence

    My teacher gave us an intuitive idea of what it means for two vectors in \mathbb{R}^2 to be linearly independent (they aren't multiples of each other) and for three vectors in \mathbb{R}^3 (they aren't on the same plane). Now the book has generalized the idea of linear independence to n...
  11. T

    Proving Linear Independence of (v,Tv,...,T^{m-1}v): A Solution

    Here's a simple question that I can't seem to get: "Suppose for some v T^{m-1}v\neq 0 and T^mv=0. Prove that (v,Tv,...,T^{m-1}v) is linearly independent." I know that m\leq \dim V and v,Tv,...,T^{m-1}v are all nonzero.
  12. K

    Cauchy-Riemann and Linear Independence

    Here's an interesting way to look at CR I feel is often overlooked: Let: z = x + i y z^{\ast} = x - i y One common form for the CR condition is to say that if some function f is analytic then it does not depend on z^{\ast}\;. That is, \frac{\partial f}{\partial z^{\ast}} = 0 But...
  13. S

    Can we prove linear independence with just matrix and vector information?

    Is it possible to prove 2 vectors are linearly independent with just the following information?: A is an nxn matrix. V1 and V2 are non-zero vectors in Rn such that A*V1=V1 and A*V2 = 2*V2. Is this enough information, or is more needed to prove the LI of the 2 vectors?
  14. D

    Need help with linear independence proof

    Hi, I don't know how to do the following proof: If (v1, ...vn) are linearly independent in V, then so is the list (v1-v2, v2-v3, ...vn-1 -vn, vn). I can do the proof if I replace 'linearly independent' with 'spans V' ...so what connection am I missing? Thanks much!
  15. mattmns

    Linear Independence of two Functions

    Hello, there is this question in the book: --------- Consider the vector space of functions defined for t>0. Show that the following pairs of functions are linearly independent. (a) t, 1/t --------- So if they are linearly independent then there are a,b in R, such that at + b/t = 0 So if we...
  16. C

    Understanding Notation and Proving Linear Independence

    Hi I just need some help on understanding some general notation in this quesiton: Prove if {x_1,x_2,..,x_m} is linearly independent then so is {x_1,x_2,...,x_i-1, x_i+1,...,x_m} for every i in {1,2,...,m}. I don't really understand what the difference between {x_1,x_2,...,x_i-1...
  17. T

    Linear Independence in R-Vector Space and Z_2

    If a,b,c are vectors in an R-vector space then their sums a+b, a+c, b+c are also linearly independent. If R is replaced by Z_2 then this fails, because there's the nontrivial solution to x(a+b)+y(a+c)+z(b+c)=0 where x=y=z=0 or x=y=z=1 right?
  18. tandoorichicken

    Linear Transformations, Span, and Independence

    Is there a linear algebra theorem or fact that says something like For a linear transformation T:Rn -> Rm and its standard m x n matrix A: (a) If the columns of A span Rn the transformation is onto. (b) If the columns of A are linearly independent the transformation is one-to-one. Is...
  19. C

    Linear independence of basis vectors

    How do I prove the linear independence of the standard basis vectors? My book is helpful by giving the definition of linear independence and a couple examples, but never once shows how to prove that they are linearly independent. I know that the list of standard basis vectors is linearly...
  20. B

    Proving Linear Independence of Non-Zero Rows in Row-Echelon Form

    Hi, can someone help me with the following question? Q. Show that if \left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to } \right\} is linearly independent and \mathop {v_{k + 1} }\limits^ \to \notin span\left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to...
  21. K

    Two questions involving linear independence

    How do you know if this: | 0_-8_5| |3_-7_4 | |-1_5_-4| | 1_-3_2| a linearly independent set? The answer at the back of the book say that it is independent, but obvious there are free variable in this matrix , thus imply a nontrival solution for AX=0, so it must be depend. Let...
  22. B

    Determining Linear Independence: Use Coordinate Vectors

    Can someone help me out with the following question? Use coordinate vectors to determine whether or not the given set is linearly independent. If it is linearly dependent, express one of the vectors as a linear combination of the others. The set S, is \left\{ {2 + x - 3\sin x + \cos x,x +...
  23. E

    Lin. transf. and linear independence

    Hi, If I transform a set of linearly independent vectors by a one-to-one linear transformation, is the transformed set also linearly independent?
  24. T

    Is Linear Independence Over Z the Same as Linear Independence Over R?

    Hey all, I need to show whether or not the following statement is true: For v_1,...,v_n\in Z^m, the set \{v_1,...,v_n\} is linearly independent over Z \Leftrightarrow it is linearly independent over R. The reverse direction is true of course, but I'm having some trouble showing whether or...
  25. G

    Determining Linear Independence: v1, v2,...vn, n≥4

    How do I determine this: Problem: The vectors: v1, v2, ... , vn, n >= 4 and are linearly independent. Determine if the following vectors are also linearly independent. a) the vectors v1 - v2, v2 + v3, v3 + v1 b) the vectors v1 - v2, 2(v2 - v3), 3(v3 - v4), ..., n(vn - v1) c) the...
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