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Questions and Answers
Consider $E = \mathbb{R}^2$. If $K = \mathbb{C}$, where $\mathbb{C}$ represents complex numbers, which of the following applications is NOT an external law?
Consider $E = \mathbb{R}^2$. If $K = \mathbb{C}$, where $\mathbb{C}$ represents complex numbers, which of the following applications is NOT an external law?
- $(\lambda, (x, y)) \rightarrow (\lambda x, \lambda y) \in \mathbb{R}^2$, where $\lambda \in \mathbb{R}$
- $(\lambda, (x, y)) \rightarrow (\lambda x, \lambda y) \in \mathbb{C}^2$, where $\lambda \in \mathbb{C}$ (correct)
- $(\lambda, (x, y)) \rightarrow (\lambda x, 0) \in \mathbb{R}^2$, where $\lambda \in \mathbb{C}$
- $(\lambda, (x, y)) \rightarrow (x, y) \in \mathbb{R}^2$, where $\lambda \in \mathbb{C}$
Which of the following statements is true regarding the properties of a vector space $E$?
Which of the following statements is true regarding the properties of a vector space $E$?
- For all $x, y \in E$, $x + y = y + x$ (addition is associative).
- For all $x, y, z \in E$, if $x + y = 0$ and $x + z = 0$, then $y \neq z$
- For all $x, y, z \in E$, $(x + y) + z = x + (y + z)$ (addition is commutative). (correct)
- For all $x, y \in E$, $x - y = y - x$ (subtraction is commutative).
If $E$ is a K-vector space and $x \in E$, what is the result of the scalar multiplication $0 \cdot x$?
If $E$ is a K-vector space and $x \in E$, what is the result of the scalar multiplication $0 \cdot x$?
- 0
- $0_E$ (correct)
- 1
- $x$
Let $E$ be a vector space. What is the significance of the element denoted as $0_E$ in the context of vector addition?
Let $E$ be a vector space. What is the significance of the element denoted as $0_E$ in the context of vector addition?
Choose the correct statement that defines an internal law on a set $E$.
Choose the correct statement that defines an internal law on a set $E$.
Given a K-vector space E, and scalars $\alpha, \beta \in K$ and a vector $x \in E$, which of the following properties is correct regarding scalar multiplication?
Given a K-vector space E, and scalars $\alpha, \beta \in K$ and a vector $x \in E$, which of the following properties is correct regarding scalar multiplication?
Which of the following operations, defined on the set of integers $\mathbb{Z}$, qualifies as an internal law?
Which of the following operations, defined on the set of integers $\mathbb{Z}$, qualifies as an internal law?
For elements in a vector space, what term is used to describe the additive inverse of an element $x \in E$?
For elements in a vector space, what term is used to describe the additive inverse of an element $x \in E$?
Given scalars $\alpha \in K$ and vectors $x, y \in E$ in a K-vector space, which of the following must hold true according to the axioms?
Given scalars $\alpha \in K$ and vectors $x, y \in E$ in a K-vector space, which of the following must hold true according to the axioms?
What distinguishes a K-vector space from a general set?
What distinguishes a K-vector space from a general set?
Flashcards
What is an internal law on a set E?
What is an internal law on a set E?
An application from E x E to E.
What is an external law on a set E with coefficients in K?
What is an external law on a set E with coefficients in K?
An application from K x E to E, where K is a set of scalars.
What defines a vector space over K?
What defines a vector space over K?
A non-empty set E with an internal law (+) and an external law (scalar multiplication) that satisfy certain axioms.
What is the neutral element for addition in a vector space called?
What is the neutral element for addition in a vector space called?
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What is the opposite of an element x in a vector space?
What is the opposite of an element x in a vector space?
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What are the elements of E and K called in a K-vector space?
What are the elements of E and K called in a K-vector space?
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What is the result of multiplying any vector by the scalar 0?
What is the result of multiplying any vector by the scalar 0?
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Study Notes
- In this chapter, K represents either the set of real numbers (R) or the set of complex numbers (C).
Vector Spaces
- A vector space over K, or a K-vector space, is a non-empty set E, equipped with an internal operation denoted by + (called addition) and an external operation denoted by . (scalar multiplication) such that certain axioms are satisfied
Definition of Internal Law
- Let E be a set. An internal law on E is an application of E x E in E
Examples of Internal Law
- Addition (+) is an internal law on Z, where (x, y) maps to x + y.
- Subtraction (-) is not an internal law on N, illustrated by 2 - 3 which is not in N.
Definition of External Law
- Let E be a set. An external law with coefficients in K is an application of K x E in E.
Examples of External Law
- If E = R^2 and K = R, the application (λ, (x, y)) -> (λx, λy) is an external law in R^2
- If E = R^2 and K = C, the application (λ, (x, y)) -> (λx, λy) is not an external law because C^2 is not included in R^2
Axioms for Vector Spaces
- The 4 axioms regarding the internal operation + that must be verified:
- E has a neutral element for +, denoted as 0_E.
- For every y in E, there exists y' in E such that y + y' = 0_E (y' is denoted as -y).
- For all x, y, z in E, (x + y) + z = x + (y + z) (+ is associative).
- For all x, y in E, x + y = y + x (+ is commutative).
- The four axioms regarding the external operation . that must be verified:
- For all x in E, 1.x = x.
- For all α, β in K and all x in E, α.(β.x) = (αβ).x.
- For all α, β in K and all x in E, (α + β).x = α.x + β.x.
- For all α in K and all x, y in E, α.(x + y) = α.x + α.y.
- Elements of E are called vectors
- The neutral element for + is called the zero vector, denoted by 0_E or simply 0.
- The opposite of x in E is denoted by -x
- Elements of K are called scalars
Properties of Vector Spaces
- For all x in E, 0.x = 0_E.
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