Purchase Solution

Vecor Spaces and Linear Combinations

Not what you're looking for?

Ask Custom Question

Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R

Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication.

1. Label the following statements as true or false

a) The zero vector is a linear combination of any nonempty set of vectors
b) If S is a subset of a vector space V, then span (S) equals the intersection of all subspaces of V that contain S..

See attached file for full problem description.

Attachments
Purchase this Solution
Solution Summary

Vecor Spaces and Linear Combinations are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Solution Preview

Please see the attached file for the complete solution.
Thanks for using BrainMass.

• Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R
• Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication.
1. Label the following statements as true or false

a) The zero vector is a linear combination of any nonempty set of vectors

True: take the linear combination with all the coefficients =0

b) If S is a subset of a vector space V, then span (S) equals the intersection of all subspaces of V that contain S..

True. It is clear that if W is a subspace containing S, then it contains span(S) ...

Purchase this Solution
Free BrainMass Quizzes
Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

View More Free Quizzes
Related BrainMass Solutions

  • General Vector Spaces

    Consider u=sin(x+theta) and v=cos(x+theta) By trigonometric identities, Then, u = sinx cos(theta) + cosx sin(theta) and v = cosx cos(theta) - sinx sin(theta) Therefore, u and v can be expressed as linear and multiplicative combinations of f and g

  • Vector space

    Show that We need 2 steps 1st step: Since u=[(u+v)+(u-v)]/2, and v= [(u+v)-(u-v)]/2, which means that u and v can be expressed as linear combinations of u+v and u-v, therefore 2nd step: It's obvious that u+v and u-v are linear combinations

  • general theorems that are usually proved in any Linear Algebra

    Now, the row space of a matrix is the set of all linear combinations of its rows, considered elements of R^m. Suppose, A has row-rank k, that is, A has k linearly independent rows, say i1, i2, ..., ik.

  • Prove ℓ(ε, F) is complete, then F is complete.

    linear spaces and Ε ≠ 0.

  • Linear Spaces, Mappings and Dimensional Spaces

    146209 Linear Spaces, Mappings and Dimensional Spaces 1) Show that if dim X = 1 and T belongs to L(X,X), there exists k in K st Tx=kx for all x in X. 2) Let U and V be finite dimensional linear spaces and S belong to L(V,W), T belong to L(U,V).

  • Closed Graph, Banach Spaces and Continuous Linear Transformations

    55886 Closed Graph, Banach Spaces and Continuous Linear Transformations Let and be Banach spaces and let . (Note: is a set of all continuous linear transformations ).

  • What is linear algebra? What is matrix algebra?

    Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations.

  • Graph of a linear function.

    (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented can't be negative quantities.) (b) What do the slope, the y-intercept, and the x-intercept of the graph represent?

  • Correalation and Regression

    Is there a linear correlation between the number of commuters and the number of parking spaces ?

View More Related Solutions