Linear independence Definition and 13 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. C

    Linear independence of Coordinate vectors as columns & rows

    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}...
  2. K

    I Trying to get a better understanding of the quotient V/U in linear algebra

    Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...
  3. brotherbobby

    Proving that the two given functions are linearly independent

    Summary:: I attach a picture of the given problem below, just before my attempt to solve it. We are required to show that ##\alpha_1 \varphi_1(t) + \alpha_2 \varphi_2(t) = 0## for some ##\alpha_1, \alpha_2 \in \mathbb{R}## is only possible when both ##\alpha_1, \alpha_2 = 0##. I don't know...
  4. GlassBones

    Isomorphisms preserve linear independence

    Homework Statement Let ##T:V \rightarrow W## be an ismorphism. Let ##\{v_1, ..., v_k\}## be a subset of V. Prove that ##\{v_1, ..., v_k\}## is a linearly independent set if and only if ##\{T(v_1), ... , T(v_2)\}## is a linearly independent set. Homework Equations The Attempt at a Solution...
  5. hilbert2

    A Linearly independent function sets

    It is well known that the set of exponential functions ##f:\mathbb{R}\rightarrow \mathbb{R}_+ : f(x)=e^{-kx}##, with ##k\in\mathbb{R}## is linearly independent. So is the set of sine functions ##f:\mathbb{R}\rightarrow [-1,1]: f(x) = \sin kx##, with ##k\in\mathbb{R}_+##. What about...
  6. M

    Linear algebra, field morphisms and linear independence

    Homework Statement Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K. Homework Equations f1, ..., fn are field morphisms => Ker (fi) = 0 (injective) The Attempt at a Solution I...
  7. LarryS

    I Measures of Linear Independence?

    My formal education in Linear Algebra was lacking, so I have been studying that subject lately, especially the subject of Linear Independence. I'm looking for functions that would qualify as measures of linear independence. Specifically, given a real-valued vector space V of finite dimension...
  8. cathal84

    I Proving a set is linearly independant

    I have two questions for you. Typically when trying to find out if a set of vectors is linearly independent i put the vectors into a matrix and do RREF and based on that i can tell if the set of vectors is linearly independent. If there is no zero rows in the RREF i can say that the vectors are...
  9. Arnab Patra

    Find general equation of x′′(t)+5x′(t)+4x(t)=0

    Suppose ##x_1(t)## and ##x_2(t)## are two linearly independent solutions of the equations: ##x'_1(t) = 3x_1(t) + 2x_2(t)## and ##x'_2(t) = x_1(t) + 2x_2(t)## where ##x'_1(t)\text{ and }x'_2(t)## denote the first derivative of functions ##x_1(t)## and ##x_2(t)## respectively with respect to...
  10. F

    Prove that three functions form a dual basis

    Homework Statement Homework Equations[/B] The Attempt at a Solution From that point, I don't know what to do. How do I prove linear independence if I have no numerical values? Thank you.
  11. kostoglotov

    Need help interpreting the Wronskian

    I'm given bases for a solution space \left \{ x,xe^x,x^2e^x \right \}. Clearly these form a basis (are linearly independent). But, unless I've made a mistake, doing the Wronskian on this yields W(x) = x^3e^x. Isn't this Wronskian equal to zero at x = 0? Isn't that a problem for...
  12. G

    Linear algebra: Find the span of a set

    Homework Statement Find the span of U=\{2,\cos x,\sin x:x\in\mathbb{R}\} (U is the subset of a space of real functions) and V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\} Homework Equations -Vector space span -Linear independence...
  13. R

    Proof: Max number of Linearly Independent Vectors

    Homework Statement Prove that a set of linearly independent vectors in Rn can have maximum n elements. So how would you prove that the maximum number of independent vectors in Rn is n? I can understand why in my head but not sure how to give a mathematical proof. I understand it in terms of...
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