Problem 21 Let \(U\) be a subspace of a vec... [FREE SOLUTION]

Chapter 6: Problem 21

Let \(U\) be a subspace of a vector space \(V\). a. If \(a \mathbf{u}\) is in \(U\) where \(a \neq 0\), show that \(\mathbf{u}\) is in \(U\). b. If \(\mathbf{u}\) and \(\mathbf{u}+\mathbf{v}\) are in \(U,\) show that \(\mathbf{v}\) is in \(U\).

Short Answer

Expert verified

a. \(\mathbf{u} \in U\) since \(U\) is closed under scalar multiplication. b. \(\mathbf{v} \in U\) since \(U\) is closed under addition.

Step by step solution

Understanding Subspaces

A subspace \(U\) of a vector space \(V\) is a subset that is closed under addition and scalar multiplication. This means if \(\mathbf{u}, \mathbf{v} \in U\), then \(\mathbf{u} + \mathbf{v} \in U\) and for any scalar \(c\), \(c \mathbf{u} \in U\).

Statement a Analysis

Given that \(a \mathbf{u} \in U\) where \(a eq 0\), we need to show \(\mathbf{u} \in U\). Since \(U\) is a subspace, it is closed under scalar multiplication.

Proof of Statement a

Since \(a \mathbf{u} \in U\) and \(U\) is a subspace, consider the scalar \(a^{-1}\) (the multiplicative inverse of \(a\)). Multiply both sides of the equation \(a \mathbf{u}\) by \(a^{-1}\), i.e., \(a^{-1}(a \mathbf{u}) = \mathbf{u}\). Hence, \(\mathbf{u} \in U\).

Statement b Analysis

Given \(\mathbf{u} \in U\) and \(\mathbf{u} + \mathbf{v} \in U\), we need to show that \(\mathbf{v} \in U\). Since \(U\) is closed under addition and scalar multiplication.

Proof of Statement b

Since \(\mathbf{u} \in U\) and \(\mathbf{u} + \mathbf{v} \in U\), subtract \(\mathbf{u}\) from \(\mathbf{u} + \mathbf{v}\): \((\mathbf{u} + \mathbf{v}) - \mathbf{u} = \mathbf{v}\). Since \(U\) is closed under addition, \(\mathbf{v} \in U\).

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Key Concepts

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

Vector Spaces

In linear algebra, vector spaces are foundational concepts that provide a framework for many mathematical theories. A vector space consists of a collection of vectors, which can be added together and multiplied by scalars, usually numbers. These operations must satisfy certain rules to ensure consistency.

For a set to be a vector space, the following conditions must hold:

Closure under addition - The sum of any two vectors in the space remains in the space.
Closure under scalar multiplication - Multiplying any vector by a scalar yields another vector in the space.
There is a zero vector that acts as an additive identity.
Each vector has an additive inverse within the space.

This structure makes vector spaces highly adaptable, applicable to physical quantities like force and velocity, and abstract concepts in fields like computer science and engineering.

Subspaces

A subspace is a special subset of a vector space that is itself a vector space under the same addition and multiplication operations. Subspaces are essential for breaking larger problems into simpler, more manageable pieces.

For a subset to be considered a subspace, it must adhere to three fundamental criteria:

The zero vector of the larger space must be in the subspace.
The subset must be closed under vector addition; adding any two vectors in the subspace results in another vector that is also in the subspace.
The subset must be closed under scalar multiplication; multiplying any vector in the subspace by a scalar results in a vector that remains within the subspace.

These properties make subspaces versatile tools for analysis, particularly in solving systems of linear equations and transforming coordinate systems.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, which is a real number. This operation is one of the building blocks in linear algebra, allowing us to scale vectors up or down.

Key properties of scalar multiplication include:

Distributivity over vector addition: For any scalar \( a \) and vectors \( \mathbf{u} \) and \( \mathbf{v} \), \( a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v} \).
Distributivity over scalar addition: For any scalars \( a \) and \( b \), and a vector \( \mathbf{u} \), \( (a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u} \).
Associative with scalar multiplication: For two scalars \( a \) and \( b \) and a vector \( \mathbf{u} \), \( (ab)\mathbf{u} = a(b\mathbf{u}) \).

Scalar multiplication is crucial for understanding how objects can be stretched or shrank, and in geometric interpretations, it changes the magnitude of the vector without altering its direction.

Vector Addition

Vector addition is the process of combining two vectors to produce a third vector. This operation is performed by adding each corresponding component of the vectors.

For example, if you have two vectors \( \mathbf{u} = (u_1, u_2, ..., u_n) \) and \( \mathbf{v} = (v_1, v_2, ..., v_n) \), their sum is given by:

\( \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, ..., u_n + v_n) \)

Vector addition follows key properties that make it compatible with operations in vector spaces:

Commutativity: \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \)
Associativity: \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \)
Existence of an additive identity, which is the zero vector, \( \mathbf{0} \), such that \( \mathbf{u} + \mathbf{0} = \mathbf{u} \)

These properties ensure that vector addition is a straightforward yet powerful operation, useful in various contexts such as finding resultants in physics or transitions in computer graphics.

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