I Subspaces, and the span of a set of vectors1. Let F be a field, and recall that F¹, the set of all n-tuples of elements of Fis a vector space over F. Let9and letW₁W₂={(a₁, a2, ..., an) ≤ F¹ | a₁ + a2 +=+ An=0}{(a₁, a2, ...,an) ¤ F” | a₁ + a2 ++ an = 1}Prove that W₁ is a subspace of Fn, but W₂ is not a subspace of Fª.

The given sets and .To prove that is a subspace of and is not a subspace of .

Answered: I Subspaces, and the span of a set of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Similar questions

Sketch each of the following sets and decide if they are open sets. Give brief reasons for your decision. (a) The open interval (-1, 1) in the normed vector space (R,| |). (b) The closed interval |-1, 1] in R. (c) The interval [0, 0) in R. (d) The line {(2x, x): a E R} in R². (e) The square (0, 1] x [0, 1) in R2. (f) The set {(0,0, 0)}U (0, 1) x (0, 1) x (0, 1) in R.
Consider the empty set, i.e. X={ }. Which property of vector space is not satisfied and how?
3. Is the set of all vectors of the form where a and b are any real numbers, a subspace W of R? Explain. If it is, find a basis for W.
2. Let P denote the vector space of all polynomials on R. Show that P is infinite- dimensional.
Determine which sets are vector spacesunder the given operations. For those that are not, list all axioms that failto hold. The set of all triples of real numbers (x, y, z) with the operations(x, y, z) + (´x, y, ´ z´) = (x + ´x, y + ´y, z + ´z) and k(x, y, z) = (0, 0, 0).
1. Let V₁, V2,...,Vn be n vectors in a vector space V. Explain what it means to say that these vectors span V.
Consider the vector space V = R 3 over the field of real numbers F = R. Do the following vectors form a basis for R 3 ? , ,
For each of the sets below, determine if the set is a vector space or subspace. a. H = {2], a, b real} а. b. V = , а — b {E 1.a, b real} С. T= 0 b. C = {the set of all complex numbers of the form a + bi, where a, b are real} e. J= {the set of all polynomials such that p(t)divides evenly by (t – 1)} [Hint: write p(t) in factored form, with the factor (t – 1) pulled out. What does the other factor look like?] d. - {the set of all odd functions: f (-x) = -f(x)} W is the set of all nxn matrices such that A² = A. h. Q is the set of all exponential functions i. f. g. S is the set of all n x n singular matrices
Let V be an n-dimensional vector space over the field of complex numbers C. Showthat V over R has dimension 2n.
Q 06: For n≥0 , the set Pn of polynomials of degree at most n consists of all polynomials of the form p(t) = a0 + a1t + a2t2 +a3t3+......+ antn form a vector space. Where the coefficients a0, a1, a2, ..... an and the variable t are real numbers.

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