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Chapter 6: Problem 48

Let \(V\) be a vector space with subspaces \(U\) and \(W\) Define the sum of \(U\) and \(W\) to be \(U+W=\\{\mathbf{u}+\mathbf{w}: \mathbf{u} \text { is in } U, \mathbf{w} \text { is in } W\\}\) (a) If \(V=\mathbb{R}^{3}, U\) is the \(x\) -axis, and \(W\) is the \(y\) -axis what is \(U+W ?\) (b) If \(U\) and \(W\) are subspaces of a vector space \(V\) prove that \(U+\mathrm{W}\) is a subspace of \(V\).

Short Answer

Expert verified
(a) The sum \( U + W \) is the \( xy \)-plane in \( \mathbb{R}^3 \). (b) \( U + W \) is a subspace of \( V \).

Step by step solution

01

Understand the Problem

We need to determine what the sum \( U + W \) is when \( U \) is the \( x \)-axis and \( W \) is the \( y \)-axis. This involves finding all possible vectors that can be formed by adding a vector from \( U \) and a vector from \( W \).
02

Identify the Vectors in U and W

Since \( U \) is the \( x \)-axis, every vector in \( U \) is of the form \((x, 0, 0)\). Similarly, since \( W \) is the \( y \)-axis, every vector in \( W \) is of the form \((0, y, 0)\).
03

Find U + W

Any vector in \( U + W \) is obtained by adding a vector from \( U \) and a vector from \( W \). Thus, any vector in \( U + W \) is \((x, 0, 0) + (0, y, 0) = (x, y, 0)\). The set of all such vectors forms the \( xy \)-plane, or \( \text{span} \{(1, 0, 0), (0, 1, 0)\} \).
04

State the Result for Part (a)

The sum \( U + W \), where \( U \) is the \( x \)-axis and \( W \) is the \( y \)-axis, is the \( xy \)-plane in \( \mathbb{R}^3 \).
05

Define a Subspace

For part (b), recall that a subset of a vector space is a subspace if it is closed under vector addition and scalar multiplication, and contains the zero vector.
06

Show Zero Vector is in U + W

The zero vector \( \mathbf{0} \) is the sum of the zero vectors in \( U \) and \( W \), i.e., \( \mathbf{0}_U + \mathbf{0}_W = \mathbf{0} \). Thus, \( \mathbf{0} \) is in \( U + W \).
07

Show Closure under Addition

Let \( \mathbf{a} = \mathbf{u_1} + \mathbf{w_1} \) and \( \mathbf{b} = \mathbf{u_2} + \mathbf{w_2} \), where \( \mathbf{u_1}, \mathbf{u_2} \in U \) and \( \mathbf{w_1}, \mathbf{w_2} \in W \). Then, \( \mathbf{a} + \mathbf{b} = (\mathbf{u_1} + \mathbf{u_2}) + (\mathbf{w_1} + \mathbf{w_2}) \). Since \( \mathbf{u_1} + \mathbf{u_2} \in U \) and \( \mathbf{w_1} + \mathbf{w_2} \in W \), \( \mathbf{a} + \mathbf{b} \in U + W \). Thus, \( U + W \) is closed under addition.
08

Show Closure under Scalar Multiplication

For \( \mathbf{a} = \mathbf{u} + \mathbf{w} \) (with \( \mathbf{u} \in U \) and \( \mathbf{w} \in W \)) and scalar \( c \), \( c\mathbf{a} = c(\mathbf{u} + \mathbf{w}) = c\mathbf{u} + c\mathbf{w} \), where \( c\mathbf{u} \in U \) and \( c\mathbf{w} \in W \). Hence, \( c\mathbf{a} \in U + W \), showing closure under scalar multiplication.
09

State the Result for Part (b)

Since \( U + W \) contains the zero vector, and is closed under addition and scalar multiplication, it is a subspace of \( V \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subspaces
In the world of linear algebra, a subspace can be understood as a smaller vector space contained within a larger one. For a set to qualify as a subspace, it must satisfy specific criteria:
  • It must contain the zero vector.
  • It must be closed under vector addition.
  • It must be closed under scalar multiplication.
If these conditions are met, then any linear combination of vectors within this set will also remain within the same set, making it a subspace. For instance, if you think of a plane within a three-dimensional space, that plane can be a subspace if it includes the zero vector and fulfills the above properties.
Vector Addition
Vector addition is a fundamental operation in linear algebra that combines two vectors to produce a third vector. When vectors are added, their corresponding components are summed up. For example, if you have vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), the result of their addition, \( \mathbf{a} + \mathbf{b} \), is \((a_1 + b_1, a_2 + b_2)\).This operation is essential in forming new elements within a vector space, especially when determining subspaces like in the provided exercise. When considering subspaces like the x-axis and y-axis of \( \mathbb{R}^3 \), any vector from the x-axis added to any vector from the y-axis forms a vector in the xy-plane.This highlights how vector addition helps build and explore vector spaces.
Scalar Multiplication
Scalar multiplication is an operation involving a vector and a scalar (a real number), resulting in a new vector. This operation scales the vector by the scalar, affecting its magnitude without altering its direction. If \( \mathbf{v} = (v_1, v_2) \) and \( c \) is a scalar, the scalar multiplication \( c\mathbf{v} \) results in the vector \( (cv_1, cv_2) \).In subspaces, scalar multiplication ensures every vector maintains its properties within the space. For example, if you take a vector from a subspace along the x-axis and multiply it by a scalar, it remains on the x-axis. This property is key in confirming that a set is indeed a subspace, as it proves that the set is closed under scalar multiplication.
Closure Property
The closure property is a significant concept when dealing with vector spaces and subspaces. It consists of two fundamental laws: closure under addition and closure under scalar multiplication.
  • Closure under Addition: This property states that the sum of any two vectors within a set must also be a vector within the same set. For example, for any vectors \( \mathbf{u_1}, \mathbf{u_2} \in U \), their sum \( \mathbf{u_1} + \mathbf{u_2} \) should be in \( U \).
  • Closure under Scalar Multiplication: Similarly, multiplying any vector in the set by a scalar must result in a vector that also belongs to that set. For instance, if \( \mathbf{v} \in V \) and \( c \) is a scalar, then \( c\mathbf{v} \) should remain in \( V \).
These closure properties confirm and ensure the integrity of a subspace, allowing us to confidently work within these sets knowing they meet these essential criteria.

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Most popular questions from this chapter

Let \(T: \mathscr{P}_{2} \rightarrow \mathscr{P}_{2}\) be a linear transformation for which \(T(1)=3-2 x, \quad T(x)=4 x-x^{2}, \quad T\left(x^{2}\right)=2+2 x^{2}\) Find \(T\left(6+x-4 x^{2}\right)\) and \(T\left(a+b x+c x^{2}\right)\)

The set of all linear transformations from a vector space \(V\) to a vector space \(W\) is denoted by \(\mathscr{L}(V, W)\). If \(S\) and \(T\) are in \(\mathscr{L}(V, W),\) we can define the sum \(S+T\) of \(\operatorname{Sand} T b y\) \\[(S+T)(\mathbf{v})=S(\mathbf{v})+T(\mathbf{v})\\] for all v in \(V\). If \(c\) is a scalar, we define the scalar multiple \(c T\) of T by c to be \\[(c T)(\mathbf{v})=c T(\mathbf{v})\\] for all \(\mathbf{v}\) in \(V\). Then \(S+T\) and \(c\) T are both transformations from \(V\) to \(W\). Prove that \(\mathscr{L}(V, W)\) is a vector space with this addition and scalar multiplication.

Find a formula for the number of invertible matrices \(\operatorname{in} M_{n n}\left(\mathbb{Z}_{p}\right) .\) [Hint: This is the same as determining the number of different bases for \(\mathbb{Z}_{p^{*}}^{n}\) (Why?) Count the number of ways to construct a basis for \(\mathbb{Z}_{p}^{n}\), one vector at a time.

Let \(S=\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n}\right\\}\) be a spanning set for a vector space \(V .\) Show that if \(\mathbf{v}_{n}\) is in \(\operatorname{span}\left(\mathbf{v}_{1}, \ldots, \mathbf{v}_{n-1}\right),\) then \(S^{\prime}=\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n-1}\right\\}\) is still a spanning set for \(V\).

Consider the subspace \(W\) of \(\mathscr{D}\), given by \(W=\operatorname{span}(\sin x, \cos x)\) (a) Show that the differential operator \(D\) maps \(W\) into itself. (b) Find the matrix of \(D\) with respect to \(\mathcal{B}=\\{\sin x, \cos x\\}\) (c) Compute the derivative of \(f(x)=3 \sin x-5 \cos x\) indirectly, using Theorem \(6.26,\) and verify that it agrees with \(f^{\prime}(x)\) as computed directly.
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