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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 38EQ

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Determine whether the given set S is a subspace of the vector space V.
GA. V = P₂, and S is the subset of P, consisting of all polynomials of the form p(z)=z² + c.
B. V is the vector space of all real-valued functions defined on the interval (a, b), and S is the subset of V consisting of those functions satisfying f(a) = f(b).
DC. V is the vector space of all real-valued functions defined on the interval (a, b), and S is the subset of V consisting of those functions satisfying f(a)=4.
DD. V=R", and S is the set of solutions to the homogeneous linear system Az-0 where A is a fixed m x n matrix.
DE.V=R¹, and S is the set of vectors of the form (0, 22, 4, 4).
OF.V= M₂ (R), and S is the subset of all upper triangular matrices.
DG. V-C(1), and S is the subset of V consisting of those functions satisfying the differential equation y) = 0.

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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?
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=Mnn,WAinMnn:detA=1
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V= F, W=finF:f(0)=1
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=F,W=finF:f(x)=f(x)
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
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 34. ,
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.
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 37. V = P, W is the set of all polynomials of degree 3
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