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Questions and Answers
In mathematics and physics, a ______ space (also called a linear space) is a set whose elements
In mathematics and physics, a ______ space (also called a linear space) is a set whose elements
vector
Scalars are often real ______, but can be complex numbers or, more generally, elements of any field.
Scalars are often real ______, but can be complex numbers or, more generally, elements of any field.
numbers
The operations of vector ______ and scalar multiplication must satisfy certain requirements, called vector axioms.
The operations of vector ______ and scalar multiplication must satisfy certain requirements, called vector axioms.
addition
The terms real vector space and complex vector space are often used to specify the nature of the ______.
The terms real vector space and complex vector space are often used to specify the nature of the ______.
Define a vector space and explain its key properties.
Define a vector space and explain its key properties.
What are scalars in the context of a vector space?
What are scalars in the context of a vector space?
What does it mean for a vector space to be a complex vector space?
What does it mean for a vector space to be a complex vector space?
What are the requirements that vector addition and scalar multiplication must satisfy in a vector space?
What are the requirements that vector addition and scalar multiplication must satisfy in a vector space?
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Study Notes
Vector Space Definition
- A vector space is a set of elements, called vectors, that can be added together and scaled by numbers called scalars.
- Scalars are often real numbers, but can also be complex numbers or elements of any field.
Vector Operations
- Vector addition and scalar multiplication are two operations that can be performed on vectors.
- These operations must satisfy certain requirements, called vector axioms.
Types of Vector Spaces
- Real vector space: a vector space where scalars are real numbers, resulting in a real coordinate space.
- Complex vector space: a vector space where scalars are complex numbers, resulting in a complex coordinate space.
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