Linear independence Definition and 175 Threads

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?

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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?

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!

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...

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...

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?

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...

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...

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...

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...

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 +...

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

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...

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...