What is Linear dependence: Definition and 67 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.
HI,
consider the 4 Maxwell's equations in microscopic/vacuum formulation as for example described here Maxwell's equations (in the following one assumes charge density ##\rho## and current density ##J## as assigned -- i.e. they are not unknowns but are given as functions of space and time...
for problem (a), all real numbers of value r will make the system linearly independent, as the system contains more vectors than entry simply by insepection.
As for problem (b), no value of r can make the system linearly dependent by insepection. I tried reducing the matrix into reduced echelon...
Summary:: x
Question:
Book's Answer:
My attempt:
The coordinate vectors of the matrices w.r.t to the standard basis of ## M_2(\mathbb{R}) ## are:
##
\lbrack A \rbrack = \begin{bmatrix}1\\2\\-3\\4\\0\\1 \end{bmatrix} , \lbrack B \rbrack = \begin{bmatrix}1\\3\\-4\\6\\5\\4 \end{bmatrix}...
I have a trouble showing proofs for matrix problems. I would like to know how
A is invertible -> det(A) not 0 -> A is linearly independent -> Column of A spans the matrix
holds for square matrix A. It would be great if you can show how one leads to another with examples! :)
Thanks for helping...
##f(x,y)##
a critical point is given by ##f_x=0## and ##f_y=0## simultaneously.
the test is:
##D=f_{xx}f_{yy}-(f_{xy})^2 ##
if ##D >0 ## and ##f_{xx} <0 ## it is a max
if ##D >0 ## and ##f_{xx} >0 ## it is a min
##D >0 ## is is a saddle
if ##D =0 ## it is inconclusive, and ##f_x## and...
Let's say we have a set of eigenvectors of a certain n-square matrix. I understand why the vectors are linearly independent if each vector belongs to a distinct eigenvalue.
However the set is comprised of subsets of vectors, where the vectors of each subset belong to the same eigenvalue. For...
Homework Statement
True or False:
If u, v, and w are linearly dependent, then au+bv+cw=0 implies at least one of the coefficients a, b, c is not zero
Homework Equations
Definition of Linear Dependence:
Vectors are linearly dependent if they are not linearly independent; that is there is an...
This is the problem:
Suppose a, b and c are linearly independent vectors. Determine whether or not the
vectors (a + b), (a - b), and (a - 2b + c) are linearly independent.
Here was my solution, which involved writing words (and hasn't actually been confirmed correct yet):
Let's align a, b and...
If we use n linearly independent vectors x1,x2...xn to form a vector space V and use another set of n linearly independent vectors y1,y2...yn to form a vector space S, is it necessary that V and S are the same? Why?
If we have a vector space Q that the dimension is n, can we say that any set of...
Homework Statement
check for linear dependecy[/B]
f(x) = cosx and g(x) = xcosx
2 functions from R to R
Homework EquationsThe Attempt at a Solution
Why this is wrong:
if i take the scalar a1 = 3, a2 = 1
i can do that since 3 is real, and a1 is in R.
so 3f(3) + -1g(3) = 0
there for we have none...
Functions f,g from R to R.
f(x) = xcosx, g(x) = cosx
for x = 0, we get f(x) = 0, g(x) = 1.
so for scalar t in R
t(f(x)) + 0 * g(x) = 0 . ==> f(x) and g(x) are linearly idepenent.
Is that right? if so in functions we search for an x that makes the function dependent?
In Andrew McInerney's book: First Steps in Differential Geometry, Theorem 2.4.3 reads as follows:https://www.physicsforums.com/attachments/5252McInerney leaves the proofs for the Theorem to the reader ...
I am having trouble formulating a proof for Part (3) of the theorem ...
Can someone help...
I feel like I almost understand the solution I've come up with, but a step in the logic is missing. I'll post the question and my solution in LaTeX form.
Paraphrasing of text question below in LaTeX. Text question can be seen in its entirety via this imgur link: http://i.imgur.com/41fvDRN.jpg...
Suppose the vectors ##v_a## and ##v_b## are linearly independent, another vector ##v_c## is linearly dependent to both ##v_a## and ##v_b##. Now if I form a new vector ##v_d##, where ##v_d = v_b+cv_c## with ##c## a constant, will ##v_d## be linearly independent to ##v_a##?
I need to check how I...
Homework Statement
Prove that if in a system of vectors: S_a =\{a_1, a_2, ..., a_n\} every vector a_i is a linear combination of a system of vectors: S_b = \{b_1, b_2, ..., b_m\}, then \mathrm{span}(S_a)\subseteq \mathrm{span}(S_b)
Homework EquationsThe Attempt at a Solution
We know due to...
Homework Statement
Prove that for a general NXN matrix, M, det(M)=0 => Linear Dependence of Columns
Homework EquationsThe Attempt at a Solution
It's not clear to me at all how to approach this. We've just started Linear algebra and this was stated without proof in lecture. I have no idea how...
Problem:
True or False? If $x$ and $y$ are linearly independent, and if $\{\textbf{x}, \textbf{y}, \textbf{z}\}$ is linearly dependent, then $\textbf{z}$ is in Span $\{\textbf{x},\textbf{y}\}$
Solution:
$\textbf{True}$. If $a\textbf{x} + b\textbf{y} = \textbf{0}$ is true and if $a\textbf{x} +...
Hi all,
I was asked by someone today to explain the notion of linear independence of a set of vectors and I would just like to check that I explained it correctly.
A set of vectors S is said to be linearly dependent if there exists distinct vectors \mathbf{v}_{1}, \ldots , \mathbf{v}_{m}...
Homework Statement
Let x1,x2,x3 be linearly dependent vectors in Rn, let A be a nonsingular n x n matrix, and let y1=Ax1, y2=Ax2, y3=Ax3. Prove that y1, y2,y3 are linearly dependent.
Homework Equations
The Attempt at a Solution
My solution was y is equal to the zero vector...
Given that
r1=2a-3b+c
r2=3a-5b+2c
r3=4a-5b+c
where a, b, c are non-zero and non coplannar vectors
How to prove that r1, r2 , r3 are linearly dependent?
I have moved with c1*r1+c2*r2+c3*r3=0
but confused how to show that at leat one of c1, c2, c3 is non-zero. We only have the information...
Homework Statement
Is the following set linearly dependent or independent? And does this set span the given space?
{eX, e-x}\inC∞(R)
Homework Equations
The Attempt at a Solution
So, if it's linearly independent, then:
k1ex +k2e-x = 0 where k1,k2=0 and only 0. But if you let k1=...
Homework Statement
For which real values of \lambda do the following vectors form a linearly dependent set in \mathbb{R}^{3}
v_{1}=(\lambda ,-\frac{1}{2},-\frac{1}{2}), v_{2}=(-\frac{1}{2},\lambda ,-\frac{1}{2}), v_{3}=(-\frac{1}{2},-\frac{1}{2},\lambda )The Attempt at a Solution
I know that...
Homework Statement
http://imgur.com/P9udvTs
Homework Equations
The Attempt at a Solution
So I set the scalar multiples of a, b, and c as x,y,z
so i had 3 equations
4x-4y+4z=0
-4x+4y-4z=0
-2x-4y-5z=0
i tried solving it numerous times each time trying to use a different...
Hello
A base of some space is a set of vectors which span the space, and are also linearly independent.
I am looking for an example of vectors which DO span some space, but are dependent and thus not a base...can anyone give me a simple example of such a case ?
thanks !
Homework Statement
Suppose V = 2^Ω where Ω = {red,blue,yellow,green}. Verify whether u={blue,green}, v={red,yellow,green}, and w={blue,yellow} are linearly independent in V
The Attempt at a Solution
I let
red = a1
blue = a2
yellow =a3
green = a4
therefore Ω = {a1, a2, a3, a4}...
Homework Statement
Suppose V = R^4 and let U = <X>, where X = {(1,0,-2,1),(2,-2,0,3),(0,2,-4,-1),(-1,2,-2,-2)}
Find linear dependence on X and use it to find a smaller generating set of U. Repeat the step until you reach a basis for U.
Homework Equations
The Attempt at a Solution...
Hi. I attached the problem and my work.
I'm not sure if I did part a) right. In the past problems I've done, they usually provide you with 3 vectors that are linearly independent, thus giving you unique values for C1, C2, C3. The matrix for this one forms:
1 1 1
0 1 3
0 0 0
Which is...
I cannot visualize the geometry for either of these ideas. Is it the case that two vectors can be linearly independent or dependent of each other? In which case, what is the dependency or independency based on? What are these two vectors independent or dependent of with respect to each other?
Homework Statement
Let S = {v_{1}, v_{2}, \cdots , v_{n}}
S is linear dependent iff at least one v in S is a linear combination of the others.
Homework Equations
The Attempt at a Solution
From here on, just take v to be a vector, and x to be some scalar please.
I really just wanted to check...
Homework Statement
Determine if vectors
<1,-1,0,2,3>
<1,0,-1,3,3>
<1,-1,0,3,0>
<0,1,-1,2,-2>
are linearly dependent or independentHomework Equations
I have been solving these questions in the book using a matrix and row reducing them. If I wound up with a free variable, I determined the...
Homework Statement
Determine if the pair of functions given are linearly independent or linearly dependent on the interval 1<x<∞, and give a reason for your answer.
y1=|x| y2=-3x
Homework Equations
I'm pretty sure this has something to do with the Wronskian.
W(f,g)=fg'-f'g
The...
Homework Statement
v1=[2 1 1 4 2]
v2=[-1 2 2 1 -1]
v3=[3 -2 1 -2 2]
v4=[4 1 4 3 3]
v5=[1 2 3 2 1]
Find if the system is linear dependent or independent. If it is dependent, express the last vector in the list (v5) as a combination of the preceding ones.
Homework Equations...
Homework Statement
Are the vectors a = [1 -1 0 1], b = [1 0 0 1] and c =
[0 -1 0 1] linearly independent?
The Attempt at a Solution
I am mainly confused about whether or not I should have my matrix in row or column form to solve this:
r 1 -1 0 1
s 1 0 0 1
t 0 -1 0 1
or
r...
Homework Statement
Suppose that E,F are sets of vectors in V with E \subseteq F. Prove that if E is linearly dependent, then so is F.
The Attempt at a SolutionRead post #2. This proof, I think, was incorrect.
If we suppose that E is linearly dependent, then we know that there exists...
I have to prove:
Consider V=F^{n}. Let \mathbf{v}\in V/\{e_{1},e_{2},...,e_{n}\}. Prove \{e_{1},e_{2},...,e_{n},\mathbf{v}\} is a linearly dependent set.
My attempts at a proof:
Since {e_{1},e_{2},...,e_{n}} is a basis, it is a linearly independent spanning set. Therefore, any vector...
I have to prove:
Let u_{1} and u_{2} be nonzero vectors in vector space U. Show that {u_{1},u_{2}} is linearly dependent iff u_{1} is a scalar multiple of u_{2} or vice-versa.
My attempt at a proof:
(\rightarrow) Let {u_{1},u_{2}} be linearly dependent. Then, \alpha_{1}u_{1}+...
Hi all,
Here is the problem:
If T: V -> W is a linear transformation and S is a linearly dependent subset of V, then prove that T(S) is linearly dependent.
Now, I know that the usual proof goes as follows:
Since S is linearly dependent, there are distinct vectors v_1, ..., v_n in S and...
Does anyone know a good way to check if a given set of vectors (assume we just know we have a set, not their values) is linearly dependent or linearly independent without a calculator?
Ex: Given a set of n-dimensional vectors, say, vector1, vector2, and vector3, how would one determine if these...
I need to prove that the set {I, A, A^2,..., A^n} is linear dependent where A is any nxn matrix. The vector space is the set of nxn matrix, considered as a nxn dimensional vector space.
Does anybody have an idea how to prove it?
Thank you very much.
Homework Statement
Show directly that the given functions are linearly dependent on the real line==>find a non-trivial linear combination of the given functions that vanishes identically.
Homework Equations
f(x) = 2x, g(x) = 3x^{2}, h(x) = 5x-8x^{2}
The Attempt at a Solution
I...
I am studying the subject of linear dependence right now and had a question on this topic. Is it possible to construct a square matrix A such that the columns of A are linearly dependent, but the columns of the transpose of A are linearly independent? My intuition tells me no, but I'm not sure...
Do all linearly dependent sets have elements that are linear combinations of each other? Or does this apply only to some of the Linearly Dependent sets?
And as a follow up question: How do you know if a set of 2x2 matrices is linearly dependent or linearly independent?
Thank you and may...
Homework Statement
Let T:Rn to Rm be a linear transformation that maps two linearly independents vectors {u,v} into a linearly dependent set {t(u),T(v)}. Show that the equation T(x)=0 has a nontrivial solution.
Homework Equations
c1u1 + c2v2 = 0 where c1,c2 = 0
T(c1u1 + c2v2) = T(0)...
I'm reading Ince on ODEs, and I'm in the section (in Chapter 5) where he talks about the Wronskian. There are quite a few things here that I don't quite understand or follow.
I'm not going to get into all the details, but briefly, suppose we have the Wronskian of k functions:
W =...
given S is a set of vectors S= (v1,v2,..vn), prove that S is linearly dependent if and only if one of the vectors in S is a linear combination of all the other vectors in S?
Can someone point me in the right direction of how to start this proof? I am completely lost.
x1= column vector (2, 1)
x2= column vector (4, 3)
x3= column vector (7, -3)
Why must x1, x2, and x3 be linearly dependent?
x1 and x2 span R^2.
The basis are these two columns vectors: (3/2, -1/2), (-2, 1)
Since x1 and x2 form the basis, x3 can be written as a linear combination of...
Homework Statement
Okay so the question is to show that these 2 functions are linearly dependent.
ie. they are not both solutions to the same 2nd order, linear, homogeneous differential equation for non zero choices of, say M, B and V
Homework Equations
f(x) = sin(Mx)
g(x) = Bx + V...