Math Problem Statement
kerjakan no 1
Solution
To solve question number 1, we are given a set of matrices where each matrix is in the form:
The problem statement suggests that the set of all matrices with real entries forms a vector space under matrix addition and scalar multiplication. Let's analyze this.
Solution:
To verify if the set forms a vector space, we need to check if it satisfies the following axioms of a vector space over the field :
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Closure under Addition: If we add two matrices of this form, the result will still be a matrix with real entries. Since , the result is a matrix in .
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Closure under Scalar Multiplication: Multiplying any matrix by a scalar will yield another matrix with real entries. Since , the result is a matrix in .
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Zero Element: The zero matrix, , is in and acts as the additive identity.
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Additive Inverse: For each matrix , there exists an additive inverse such that their sum is the zero matrix.
Since these properties are satisfied, we can conclude that the set of all matrices with real entries, , indeed forms a vector space under matrix addition and scalar multiplication.
Would you like further details or have any questions?
Follow-Up Questions:
- What other types of matrices could form a vector space?
- How does matrix addition and scalar multiplication work in higher dimensions?
- Can we find a basis for and determine its dimension?
- What is the role of the zero matrix in vector spaces?
- How can we verify vector space properties in more abstract sets?
Tip:
Always verify the basic properties (closure, identity, inverse, and distributive properties) to confirm if a set with defined operations forms a vector space.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Spaces
Matrix Operations
Formulas
Matrix addition
Scalar multiplication for matrices
Theorems
Vector space axioms
Closure under addition
Closure under scalar multiplication
Suitable Grade Level
Undergraduate - Linear Algebra
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