Let W1 and W½ be two subspaces of a finite dimensional vector space V over afield F.1. Show that Win W2 is a vector subspace of V, but that Wi U W2 need notbe a vector subspace in general.2. Show thatdim(W1) + dim(W2) – dim(W1 n W2) = dim(W1 + W2),where W1 + W2 just denotes the span of W1 U W2 in V. (Hint: Apply therank nullity theorem to the natural map W1 O W2 → V.)3. (Cultural comment, for extra credit.) The formula of part 2 is reminiscent ofthe simple combinatorial formula#S1 + #S2 – #(S1 n S2) = #(SI U S2).How might you deduce t from the formula of part 2? (f course this is a bitperverse since (3) is arguably easier than (2), but it illustrates the importantway in which linear algebra resonates with combinatorics.)

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Answered: 1. Show that WW10 W2 is a vector…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 41CR: Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the...

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Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many vectors are in a basis for the row space of A? c How many vectors are in a basis for the column space of A? d Which vector space Rk has the row space as a subspace? e Which vector space Rk has the column space as a subspace?
Repeat Exercise 41 for B={(1,2,2),(1,0,0)} and x=(3,4,4). Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
In Exercises 1-4, let S be the collection of vectors in [xy]in2 that satisfy the given property. In each case either prove that S forms a subspace of 2 or give a counterexample to show that it does not. xy0
Consider the vector spaces P0,P1,P2,...,Pn where Pk is the set of all polynomials of degree less than or equal to k, with standard operations. Show that if jk, then Pj is the subspace of Pk.
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1. Show that WW10 W2 is a vector subspace of V, but that W1 U W2 need not be a vector subspace in general. 2. Show that