Linearly Definition and 217 Threads

I have a problem that's a bit more complicated than this, but the basic idea is this: a force is changing over time, it changes linearly. And I need to find the velocity. after multiple changes over time. ------------------------ given: velocity, v0 = 3 m/s mass, m = 3 kg. at t = 0, F =...

I've looked it up and can't really find a clear answer.. For a system of 3 simultaneous linear equations, is there any difference between 'the equations are linearly independent' and 'the equations have a unique solution'. If so what is it? Thanks!

Homework Statement For Any 5 x 3 matrix A, explain why rows of A must be linearly dependent. The Attempt at a Solution No idea please drop a hint

Hi, first off - Go easy on me, I'm only learning :redface: My book is talking about linear independence. As I understand the concept it means that a vector has to be in a different direction, i.e. non-collinear, with the vector in comparison. Mathematically, my book has defined...

Homework Statement When a does a capacitor increase in charge linearly? Homework Equations C=Q/V The Attempt at a Solution If I am correct, it is when the capacitor is being directly charged by a battery. When there is a resistor, we get an RC circuit scenario where capacitor...

Homework Statement Suppose that A, B, and C are not linearly independent. Then show how the \alpha_{i} can be computed, up to a common factor, from the scalar products of these vectors with each other. Hint: Suppose that there are non-zero values of the \alpha_{i}'s that satisfy...

Homework Statement So I'm trying to find a basis for the space that is spanned by the given vectors. {(1,0,0,1) (-2,1,-1,1) (6,-1,2,-1) (5,-3,3,-4) (0,3,-1,1)} These are written as column vectors. Homework Equations None really (that I know of) The Attempt at a Solution So I...

Let x1, x2,...,xk be linear independent vectors in a vector space V. If we delete a vector, say xk, from the collection, will we still have a linearly independent collection of vectors? Explain. By deleting a vector from linearly independent span, the other vectors, I believe, will remain...

Let x1, x2,...,xk be linear independent vectors in a vector space V. If we delete a vector, say xk, from the collection, will we still have a linearly independent collection of vectors? Explain. By deleting a vector from linearly independent span, the other vectors will remain independent...

For the use of linear and logarithmic interpolation, how can I produce linearly and logarithmically spaced arrays in C? I know in Matlab the code is A = linspace(a,b,c) or A = logspace(a,b,c) but how can I do this in C?

Homework Statement Determine whether the pairs of functions are linear dependent or linearly independent. f(x) = x^3 g(x) = x2|x| Homework Equations The Attempt at a Solution g(x)=x^2|x| = x^2*sqrt[x^2] = sqrt[x^6] = x^3 f'=3x^2 g'=3x^2 fg'-f'g = 0 Linearly Dependent...

If A is a 3x3 Matrix and {v1, v2, v3} is a linearly dependent set of vectors in R^3, then {Av1, Av2, Av3} is also a linearly dependent set? Is this true? Can someone please explain why or why not?? What I think: I think it is true because I read that a linear transformation preserves the...

Standard Matrix for R^3 -> R^4 where if domain is Linearly Independent so is codomain Homework Statement 5. Suppose T : R^3 -> R^4 is a linear transformation with the following property: For any linearly independent vectors v_1, v_2 and v_3 in R^3, the images T(v_1), T(v_2) and T(v_3) are...

I have a quick regarding a definition for linear dependence that my professor gave in class... A set of vectors {v_{1},v_{2},...v_{k}}, are considered linearly dependent provided there are scalars c_{1},c_{2},...c_{k} that are not all zero, such that c_{1}v_{1} + c_{2}v_{2} + ... c_{k}v_{k} =...

Homework Statement Show that if vectors v1 , . . . , vk in a vector space V have the properties that v1 does not = 0, and each vi is not in the span of the preceding ones, then the vectors are linearly independent. Conversely, show that if v1 , . . . , vk is an ordered list of linearly...

Homework Statement In the space of 2 by 2 matrices, find a basis for the subspace of matrices whose row sums and column sums are all equal. (Extra credit: Find five linearly independent 3 by 3 matrices with this property) The Attempt at a Solution The first one is ok. The matrix is...

Homework Statement If S = {v1, v2, v3} where v1, v2 and v3 are in \mathbb{R}^2, then the vectors v1, v2 and v3 are linearly independent. Homework Equations NoneThe Attempt at a Solution I thought the answer was true, but I know the correct answer is false and I'm not sure why. Here was my...

Hi all, I have been studying Linear Algebra for an upcoming exam, and one question has puzzled me slightly! How do you determine of a vector in R4 is linearly independent? Given three vectors, each with 4 rows, I know you are meant to arrange them into a matrix, like this: \[ \left(...

Homework Statement Suppose that {v1,...,vn} is a linearly independent set in a vector space V and let L:V --> W be a one-to-one linear mapping. Prove that {L(v1),...,L(vn)} is linearly independent. Homework Equations None The Attempt at a Solution If L is a one-to-one linear...

I know that the basis is finite set of vectors that are linearly independant and SPANS that set. But why is when you find the basis for the row space for example the answer is {[u1,u2,...,um}} and not span{[u1,u2,...,um}]. I got this wrong in a test. I don't see why eventhough the definition...

I'm struggling to get my mind around this concept: If you have a spring with a known spring constant, and you put a specific mass on it, compress it a certain distance and then release it, can you predict the distance it will travel with the information supplied? Or in other words, can I...

Is it possible to find out the relative concentration of c02 by looking at the emission or absorption from a hot gas, T = 1000K, containing among other C02? What I want to find out is if the emission depends linearly on the concentration. (in the IR region) What theoretically equations may I...

Homework Statement if linearly polarized light has an electric field strength E, what is its field strength after it has passed through polarizing material that has a transmission axis oriented at an angle theta relative to the direction of E? Homework Equations none.The Attempt at a Solution...

Homework Statement Let A = \left[\begin{array}{ccccc} 2&0&-2&1&-3 \\ 1&1&-3&0&-2\\1&0&-1&-1&3 \end{array}\right] a) Find two linearly independent vectors u and v in R5 such that span{u,v}={x \in R^5 : Ax=0}. b) Find three linearly independent vectors u,v and w such that col(A) =...

Homework Statement Prove that two vectors are linearly dependent if and only if one is a scalar multiple of the other. Homework Equations The Attempt at a Solution This seems at glance to be a fairly easy proof: Part I Assume that vectors u and v are linearly dependent...

Homework Statement If u1 and u2, u2 and u3, u1 and u3 are Linearly Independent, does it follow that {u1,u2,u3} is Linearly Independent?

If au + Bv + yw = a'u +B'v+y'w and a=a', then u,v,w are linearly dependent

Homework Statement Let u_1...u_n be linearly independent column vectors in R^n and A an invertible n x n matrix. Prove that the vectors Au_1...Au_n are linearly independent. Homework Equations The Attempt at a Solution It is easy to prove this using scalars and the definition of...

Homework Statement When are the vectors (1,x) and (1,y) linearly independent in R^2? When are the vectors (1,x,x^2), (1,y,y^2), and (1,z,z^2) linearly independent? Generalize to R^n. The Attempt at a Solution At first, I think I misconstrued the question and I ended up finding basis...

Homework Statement Let V = {differentiable f:R -> R}, a vector space over R. Take g1,g2,g3 in V where g1(x) = e^{}x, g2(x) = e^{}2x and g3(x) = e^{}3x. Show that g1, g2 and g3 are distinct.Homework Equations If g1-g3 are linearly independent, it means that for any constant, k in F (field) then...

Homework Statement Given vectors: a1 = (1; 2; 0), a2 = (2; 1; 3), a3 = (0; 3; -3). Find out if these vectors are linearly independent. Homework Equations The Attempt at a Solution \begin{cases}...

If you have a 3x3 matrix AB of rank 2, and you are asked to find two linearly independent vectors in the column space of A, are those the same vectors in the column space of AB? I figure it is since if the vectors are linearly independent they wouldn't change. Am I right?

Hi guys, I am solving a problem in the form: (ATx=0 where A is a matrix of known numbers and I am solving for x. After performing reduction and multiplying ATx, I am left with the following equations: -X1 + X4 - X5 = 0 -X2 + X4 = 0 -X3 + X4 -X5 + 28X6 = 0 From these equations, I...

So the question is... Let v1, v2,...,,vn be linearly independent vectors in a vector space V. Show that v2,...,vn cannot span V. I honestly have found myself completely lost lately and I suck at writing proofs. So this is what I see, v1, v2,..., vn is linearly independent iff...

Homework Statement Show that the set of functions: x^2 3x+2 x-1 2x+5 are linearly dependent. Homework Equations - The Attempt at a Solution I know that you have to show that you can put constants in front of each equation (that aren't all zero) such that: c1y1 + c2y2...

Homework Statement Suppose v_1,...,v_k is a linearly dependent set. Then show that one of the vectors must be a linear combination of the others. Homework Equations . The Attempt at a Solution I have attached an attempt at the problem. Thank you for help

Hi I am supposed to, without calculation, find 2 linearly independent eigenvectors and a eigenvalue of the following matrix A 5 5 5 5 5 5 5 5 5 The eigenvalue is easy -- it is 15. And I can find one eigenvector, [1 1 1] (written vertically), but another without calculation? Is there...

suppose that \{ v_1,v_2,v_3\} is a set of dependent vectors. Prove that \{v_1,(v_1+v_2),v_3\} is also a set of dependent vectors? Ok, I have two ways of going about it. My concern is about the first one, it looks like it is not correct the way i did it, i think so, because i managed to find...

1. Homework Statement Find two linearly independent solutions valid for x>0 xy''+y=02. Homework Equations well first of all, all the solutions that i get to see in the forums is interms of the euler formula.. but in the course that I am taking(advanced engineering mathematics), it is...

Hi everyone, I was reading about Gram-Schmidt process of converting two linearly independent vectors into orthogonal basis. But, as I understand, if two vectors are linearly independent then they are orthogonal! isn't that right?? Could anybody explain... please.

Hi wondering if can anyone help me... I've gotten so bogged down in all the rules and stuff for singular/non-singular matrices I've completely confused myself! Can anyone tell me is it true to say that if I have a matrix P, det(P) is NOT EQUAL to 0, then the vectors that would form the...

Homework Statement Determine whether the given vector functions are linearly dependent or independent on the interval (-\infty, \infty) \begin{pmatrix} t \\ 3 \end{pmatrix}, \begin{pmatrix} 4 \\ 1 \end{pmatrix}Homework Equations The Attempt at a Solution I computed the wronskian to be t-12...

Homework Statement Given x_{1}= \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}, x_{2}= \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}, x_{3}= \begin{pmatrix} 2 \\ 6 \\ 4 \end{pmatrix} Find the dimension of Span(x_{1}, x_{2}, x_{3}) Homework Equations If V is a vector space...

Say f is a continuous function on R. How could I find two linearly independent solutions of (y' + f(x)y)' = 0? Notice that there is no hypothesis about f being differentiable, so the obvious method of attack (taking the derivative of each term in the parenthesis and working off the resultant...

I have a final tomorrow. Can anyone guide my through this proof? I know i have to wirite the set of vectors as a linear combo, but what can I do nect? Prove that any set of vectors containing the zero vector is linearly dependent.

Homework Statement I have three vectors in R^(2x2): (1 0 , 0 1) (That is "1 0" horizontal first line, and "0 1" horizontal second line), (0 1, 0 0) and (0 0, 1 0). I have to determine if they are linear independent or not. I know how to do it in R^(2x1), but not in R^(2x2). What's the...

I have to prove that any set of vectors containing the zero vector is linearly dependent. How can I approach this?

Homework Statement Show that the given set is linearly dependent and write one of the vectors as a linear combination of the remaining vectors. {(1,2,1,0), (3,-4,5,6), (2,-1,3,3), (-2,6,-4,-6) } Homework Equations The Attempt at a Solution I've tried setting up equations like (0,0,0,0) = c1 *...

Homework Statement Is {(1,4,-6), (1,5,8), (2,1,1), (0,1,0)} a linearly independent subset of R^3. Justify your answerHomework Equations The Attempt at a SolutionI asssumed a(1,4,-6) + b(1,5,8) + c(2,1,1) + d(0,1,0) = 0 then i set up the system a + b + 2c = 0 4a + 5b + c + d = 0 -6a + 8b + c...

Are the functions y=0 and y=sinh(pi*x) linearly dependent or linearly independent on the intercal x>0? I'm not sure what I'm supposed to do here, but I try to divide them: 0/sinh(pi*x). This is certainly 0, since sinh (pi*x) is positive when x>0. Since 0 is a constant, the functions must...