What is Linear dependence: Definition and 66 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. P

    Linear Dependence of Equation Vectors

    Homework Statement Determine whether the members of the given set of vectors are linearly independent for -\infty < t < \infty. If they are linearly dependent, find the linear relation among them. x(1)(t) = (e-t, 2e-t), x(2)(t) = (e-t, e-t), x(3)(t) = (3e-t, 0) (the vectors are written as...
  2. S

    About the linear dependence of linear operators

    Notations: F denotes a field V denotes a vector space over F L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is composition of functions. τ denotes a linear operator contained in L(V) ι...
  3. M

    Proving Linear Dependence of Trig Functions on the Real Line

    Homework Statement show {cos x ,sin x , cos 2x , sin 2x , (cos x − sin x)^2 − 2*sin^2( x)} is not a linearly independent set of real valued functions on the real line R. The Attempt at a Solution Not linearly independent = linearly dependent? So if f(x) = cos (x) g(x) = sin (x) m(x)...
  4. A

    Linear Dependence Check: [-1, 1]

    Check for Linear Dependence for: \sin \pi x [-1, 1] I'm thinking it's Linear Dependent. Since it says that any linear combination must be 0. a*x + b*y = 0, a = b = 0. So for any integer x, the value is 0. So [-1, 1] works.
  5. L

    Exploring Linear Dependence in $\mathbb{Z}_2^3$

    HI. Okay Consider \[\left(\begin{array}{c} 0\\ 0\\ 0\end{array}\right)\] , \[\left(\begin{array}{c} 1\\ 1\\ 0\end{array}\right)\] , \[\left(\begin{array}{c} 1\\ 0\\ 1\end{array}\right)\] , \[\left(\begin{array}{c} 0\\ 1\\ 1\end{array}\right)\] as a subspace of...
  6. F

    Linear Dependence: Is (1,0,0), (-3,7,0) & (1,1,0) Independent?

    (1,0,0) (-3,7,0) and (1,1,0) I'm trying to work out if these vectors are linearly independent or not. Intuitively i believe they are dependent as they span the xy-plane.. but then how do i work out the linear combinations. e.g: (1,0,0) = a(-3,7,0) + b(1,1,0) where a and b are real...
  7. M

    What Values of x Make These Vectors Linearly Dependent?

    Homework Statement For what values of x are the vectors, [1] [x] [2x] [1] [-1] [-2] [2] [1] [x] linearly dependent? Homework Equations The Attempt at a Solution I made a matrix, [1 ; 1 ; 2 ; 0] [x ;-1 ; 1 ; 0] [2x;-2 ; x ; 0] but I'm having trouble figuring...
  8. D

    Linear dependence and inner product space

    Homework Statement The following is from the book Linear Algebra 3rd Edn by Stephen Friedberg, et al: Here aj are scalars of field F and vj are vectors of inner product space V. Homework Equations Theorem 6.3: The Attempt at a Solution Now I don't understand why theorem 6.3...
  9. E

    Linear Dependence: Complex Equations & Conjugates

    Hi All A complex equation and its complex conjugate are linearly dependent or independent thanks eman
  10. S

    Linear Algebra: Linear dependence of vectors?

    Homework Statement Given: v_1 = \left(\begin{array}{cc}1\\-5\\-3\end{array}\right) v_2 = \left(\begin{array}{cc}-2\\10\\6\end{array}\right) v_3 = \left(\begin{array}{cc}2\\-9\\h\end{array}\right) For what value of h is v_3 in Span{v_1, v_2} and for what value of h is...
  11. N

    Understanding Linear Dependence in ODEs: Exploring the Not All Zero Condition

    Now I am reading over a theorem, which is very easy to understand, except for a small caveat. Bascally: A set of functions are said to be linearly dependent on an interval I if there exists constants, c1, c2...cn, not all zero, such that c1f1(x) + c2f2(x) ... + cnfn(x) = 0 Well the...
  12. A

    Linear dependence of bessel equation

    why for bessel equations, if n isn't an integer, you can have the solution y(x)=(c1)Jn(x) +(c2)J(-n)x but isn't true if n's an integer?
  13. M

    Solving Linear Dependence of a Matrix

    Hello everyone, I'm finishing up some matrices review and im' confused on this question i have the matrix: -1 -3 -1 2 5 13 3 -8 3 10 9 -8 1 4 7 -4 I row reduced got this: 1 0 0 3/5 0 1 0 -4/5 0 0 1 -1/5 0 0 0 0 So you can see that this isn't a basis due to column 5 not being 0 0 0...
  14. tandoorichicken

    Is it possible to prove linear dependence using a simple approach?

    This may be a really simple proof but its giving me grief. If {v_1, v_2, v_3} is a linearly dependent set of vectors in \mathbb{R}^n, show that {v_1, v_2, v_3, v_4} is also linearly dependent, where v_4 is any other vector in \mathbb{R}^n. Any hints on where to start? I started out by writing...
  15. G

    Proving Linear Dependence in Pm(F) Using Polynomials with p_j(2)=0

    Suppose that p_0,p_1,p_2...,p_m are polynomials in Pm(F) such that p_j(2)=0 for each j. Prove that (p_0,...,p_m) is not linearly independent in Pm(F). So far I have, suppose that there is a polynomial in the list that is of degree 0, then that polynomial must be 0, hence the list is...
  16. V

    How Can You Prove Linear Dependence in Vector Sets?

    I've already found the answer to this solution but I want to check my methods because the class is very proof-based and the professor likes to take off points for style in proofs on tests: 5. Is {(1, 4, -6), (1, 5, 8), (2, 1, 1), (0, 1, 0)} a linearly independent subset of R^3? Justify your...
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