Linear independence Definition and 175 Threads

Can i use the invertability of a matrix as an alternative way of determining the linear independence of a set? Thank you.

What could we say if a matrix is invertible? Could we say that it can span and is linearly independent?

Hello everyone. I was going through my Linear Algebra Done Right textbook that threw me off. I hope this forum is appropriate for my inquiry; while there is no problem I'm trying to solve here, I don't know whether just asking for clarification would belong to the homework forum instead. If...

Homework Statement I have five sets of three functions that I need to test for linear independence. I know that I could write them as a linear combination of scalar multiples of the functions, set it equal to 0 and see if I can solve for the scalar multiples. To clarify I mean: Functions...

Homework Statement "if u,v,w are linearly independent then au+bv+cw=0 for some a,b,c in R" first of all is this correct? and how do i proof if this is correct? should i show like this, i know 0u+0v+0w=0 so that imply that statement is true? because 0 is in R is this...

How can I prove given an arbitrary set of vectors v1 and v2, given they are linearly independent, that their sum (v1 + v2) is also linearly independent?

Homework Statement Hi I seem to remember that if you have a homogenous ODE y'' + p(t)y' + q(t)y = 0 which have the solutions y1 and y2. Where we are told that y1(t) \neq 0 then y1 and y2 are linear independent. I found the simular claim on sosmath.com but are they simply...

Homework Statement If a set is linear independent, proove that every of its non empty subset is linear independent. i'm sorry, but I'm not sure is my sentence correct or not, Homework Equations n/a The Attempt at a Solution let {v1,v2,v3,...,vn} be linear independent set. so...

How do you know when a matrix (or equivocally a system of equations) is linearly independent? How do you know that it's linearly dependent? For example, given this matrix, [ 1 1 2 1] [-2 1 4 0] [ 0 3 2 2] How do we know if this matrix is linearly independent or dependent...

If x1, x2,..., xn span \mathbb{R}^n, then they are linearly independent. This is true since n-1 vectors can't span R^n. How can this be written in a more meaningful way?

(Note: this isn't a homework question, I'm reviewing and I think the textbook is wrong.) I'm working through the Gram-Schmidt process in my textbook, and at the end of every chapter it starts the problem set with a series of true or false questions. One statement is: -Every orthogonal set...

Homework Statement A is a 3x3 matrix with distinct eigenvalues lambda(1), lambda(2), lambda(3) and corresponding eigenvectors u1,u2, u3. Suppose you already know that {u1, u2} is linearly independent. Prove that {u1, u2, u3} is linearly independent. Homework Equations ?? The...

Homework Statement Define linear functionals lk on Pn by lk(p) = p(tk) Show that the lk are linearly independent. Homework Equations q1(t)= (t-t2)(t-t3)...(t-tn) The Attempt at a Solution I really don't know where to start. I do have the following equation: If the lks are...

Homework Statement Let u,v,w be three linearly independent vectors in ℝ7. Determine a value of k, k= , so that the set S={u-3v,v-5w,w-ku} is linearly dependent. Homework Equations The Attempt at a Solution I don't really know why knowing that we're in ℝ7 will help. I know a...

Homework Statement I have a set of Vector v_1,v_2,v_3,v_4 in \mathbb{R}^4 and need to show that E = v_1,v_2,v_3,v_4 is an ordered basis for \mathbb{R}^4 The Attempt at a Solution I know that for this being the case v = c_1 \cdot v_1 + \cdots + c_4\cdot v_4 where v \in...

Can a non square matrix have linearly independent columns? I can't take the determinant so I can't tell.

Hi I just want to confirm something. I have A,B\in M_{nm}(\mathbb{C}) and I want to prove they are linearly independent. Since they are over the complex field, do I have to prove: (a_1+ib_1)A+(a_2+ib_2)B=0 \ \Leftrightarrow \ a_1=a_2=b_1=b_2=0 ? Thanks.

Homework Statement Consider the two lines, given in the paramentric form L1: x = (0, 1 ,2) + s(1, 0, 2) L2: x = (4, 2, c) + t(-2, 0, d) where c and d are constants. a) For what value of d are the lines parallel? b) With the value of d above, for what value(s) of c (if any) are the...

If u,v andw are three linearly independent vectors of some vectorial space V, show that u + v , u-v and u -2v + w are also linearly independent. Okay, first of all, i know that: \lambda_{1} \times u + \lambda_{2} \times v + \lambda_{3} \times w = (0,0,0) admits only the solution that...

Homework Statement If the set {v1,v2,v3} of vectors in R^(m) is linearly dependent, then argue that the set {v1,v2,v3,v4} is also linearly dependent for every choice of v4 in R^(m). Homework Equations Definitions would be more relevant so... Linearly Independent: If the only solution...

suppose v1,...,vk are nonzero vectors with the property that vi.vj=0 whenever i is not equal to j. Prove that {v1,...,vk} is linearly independent.

Let V be a linear space and u, v, w \in V. Show that if {u, v, w} is linearly independent then so is the set {u, u+v, u+v+w}

This is not really a homework problem but it relates to a number of problems, so I thought this would be the most appropriate place to post it. Homework Statement The basic question is about how we define linear dependence in a vector space. For a vector space over some field \mathbb{F}, we...

Homework Statement If V1...V4 are linearly independent vectors in R4, then {V1, V2, V3} are also linearly independent. True or False. The Attempt at a Solution My solution involved reducing the problem down the 3 vectors in R3. Then show a counter example of this in R3 although I...

Show that the set of functions SIN(nx) where n=1,2,3... is linearly independent ?

Homework Statement Let A and B be vector spaces, T:A->B be a linear transformation. Give examples of: (a) T, where a(1),... a(n) are linearly independent vectors in A, but T(a(1)),...T(a(n)) are not. (b) T, where T(a(1)),...T(a(n)) span the range of T, but a(1),... a(n) do not span A...

Homework Statement Prove that if u and v are given non-zero vectors in the arbitrary inner-product space V, and are such that <u,v>=0, then {u,v} is a linearly independent subset of V. Homework EquationsThe Attempt at a Solution I have no idea where to start. It's hard to prove because the...

CAN somebody, please write down a formula defining the linear independence of the following functions?? { e^x,e^{2x}}

How can you show that an arbitrary n \times n matrix has n linearly independent eigenvectors? What if all you know about the matrix is that it's the product of a positive-definite matrix and a semi-positive-definite matrix?

Homework Statement If linearly dependent, write one matrix as a linear combination of the rest. \left[\begin{array}{cc} 1&1 \\ 2&1 \end{array}\right] \left[\begin{array}{cc} 1&0 \\ 0&2 \end{array}\right] \left[\begin{array}{cc} 0&3 \\ 2&1 \end{array}\right] \left[\begin{array}{cc} 4&6 \\ 8&6...

Homework Statement x1, x2, and x3 are linearly dependent. Show that x1 and x2 are linearly independent. Homework Equations After reduction using gaussian elimination, x1, x2, and x3 are proven to be linearly dependent because x1 and x2 are defined by x3 (being the free variable) as...

I attempted the proof but I don't know how to complete it.. Let u,v,w be linearly independent vectors and x is in <u,v,w>. Then there are unique a,b,y such that x=au+bv+yw

Linear independence!? Homework Statement Let {p, q} be linearly independent polynomials. Show that {p, q, pq} is linearly independent if and only if deg(p)>=1 and deg(q)>=1. The Attempt at a Solution I am pretty sure the statement to prove is incorrect. If we use deg(p) = -1 and...

Homework Statement Let u and v be two nonzero vectors in R^2. If there is no c E R such that u = cv, show that {u, Bv} is a basis of R^2 and that R^2 is a direct sum of the subspaces generated by U = <u> and V = <v> respectively. Homework Equations Clearly, u and v are linearly...

Determine whether the set {[1,2,-1,6], [3,8,9,10],[2,-1,2,-2]} is linearly independent. 3. The Attempt at a Solution I construct A = \left[\begin{array}{ccccc} 1 & 2 & -1 & 6 \\ 3 & 8 & 9 & 10 \\ 2 & -1 & 2 & -2 \end{array}\right] The row echelon form is A =...

Please refresh my memory; if a finite set S is L.I., then does this imply the existence of a set T of the same size (i.e. |T| = |S|) so that the elements T are pairwise orthogonal?

Homework Statement A=[-2 -7 -1] B=[-2 -4 -3] C=[0 6 -4] Determine whether or not the three vectors listed above are linearly independent or linearly dependent. I have determined that they are linearly DEPENDENT. If they are linearly dependent, determine a non-trivial linear...

Can you help me with this, or at least give me an idea how to proceed: Let T:V->V be a linear operator on the vector space over the field F. Let v is in V and let m be a positive integer for which v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0, but T^m(v) is equal...

Homework Statement For each n \in \mathbb{N}, let f_n(x) = e^{nx} for x \in \mathbb{R}. Prove that f_1, ... , f_n are linearly independent vectors in {\cal F}(\mathbb{R}, \mathbb{R}) Homework Equations The Attempt at a Solution I know that the simple way to prove this for n=2...

My professor says that a linearly independent subset of a vector space automatically spans the vector space, and that a subset of a vector space that spans the vector space is automatically linearly independent. I don't understand why either of these is true.

Homework Statement Let v, x_1, x_2, x_3 be elements in R^4, and suppose that there are distinct real numbers c1, c2, and c3 such that v = c_1*x_1 + c_2*x_2 + c_2*x_2. Prove that x_1, x_2, and x_3 are independent. The Attempt at a Solution Let A=[x_1 x_2 x_3]. Then Col(A) = span{x_1...

the vectors: v= [-5, -8, 7], u= [2, 4, (-17+k)] and w= [2, 7, 1] are linearly independent if and only if k does not equal ___? - note that the vectors are supposed to be setup vertically with only one column and 3 rows. det[v, u, w] The Attempt at a Solution - I tried...

I'm kinda confused about whether the vectors in a linear span has to be independent. It makes sense intuitively. For example say v and u spans a plane. Then v and u has to be linearly independent. Otherwise they would lie in a line. Can anyone give me an example where vectors span a space and...

Can I ask for some help? Suppose that {v1,v2...vn} is a linearly independent set of vectors and A is a singular matrix. Prove or disprove: The set {Av1, Av2, ...Avn} is linearly independent.

I can't seem to figure this one out: Question: Let D be a nonempty subset of a vector space V over a field F. Let B be a finite linearly independet subset of span D having n elements. Prove there exists a subset D' of D also having n elements such that span[(D-D') U B] = span(D)...

I need some help with examples. Especially number 2. 1) Name a subset which is closed under vector addition and additive inverses but is not a subspace of R squared. I think I got this one. {(x,y) st x,y are elements of integers} because this isn't closed under scalar multiplication...

I think I am missing a key info below. I have listed the problem statement, how I am approaching and why I think I am missing something. Please advise why I am wrong. Thanks Asif ============ Problem statement: Let T: U->V be an isomorphism. Let U1, U2,...,Un be linearly...

Homework Statement Using the wronskian (determinant basically), determine if e^x, sin(x), cos(x) are linearly independent Homework Equations I used this: | e^{x} sin(x) \:cos(x)| |e^{x} cos(x) -sin(x)| |e^{x} -sin(x) -cos(x)| But pretend that's just a 3x3 matrix and you take...

Homework Statement Proof 1: Show that S= {v1, v2, ... vp} is a linearly independent set iff Ax = 0 has only the trivial solution, where the columns of A are composed of the vectors in S. Be sure to state the relationship of the vector x to the vectors in S 2. The attempt at a solution As far...

Homework Statement Let V and W be vector spaces, Let T: V --> W be linear, and let {w1, w2,..., wk} be linearly independent subset of R(T). Prove that if S = {v1,v2,...vk} is chosen so that T(vi) = wi, for i = 1, 2,...,k, then S is linearly independentHomework Equations The Attempt at a...