Math Problem Statement
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Solution
The question asks about the set of all matrices with real entries, denoted as , and states that under matrix addition and scalar multiplication, this set forms a vector space over the field of real numbers.
To analyze this, we need to verify that satisfies the properties of a vector space over .
Solution:
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Vector Space Properties: A set forms a vector space over a field if it satisfies the following conditions with respect to addition and scalar multiplication:
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Closure under addition: For any two matrices and , their sum is also a matrix with real entries, so the set is closed under addition.
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Associative property of addition: Matrix addition is associative, meaning for any matrices , , and , we have .
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Existence of an additive identity: The zero matrix serves as the additive identity, as for any matrix .
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Existence of additive inverses: For every matrix , there exists an additive inverse such that .
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Closure under scalar multiplication: For any scalar and matrix , the product is also a matrix with real entries.
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Distributive properties: Scalar multiplication is distributive over matrix addition and scalar addition:
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Compatibility of scalar multiplication with field multiplication: For any scalars and any matrix , .
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Existence of a multiplicative identity: The scalar acts as a multiplicative identity, as for any matrix .
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Conclusion: Since satisfies all these properties, it is indeed a vector space over the real numbers .
Would you like more detailed explanations for any of these properties?
Related Questions:
- What is the dimension of as a vector space over ?
- How can we determine a basis for ?
- Can be extended to form a different kind of algebraic structure, such as a ring or algebra?
- What is the role of the identity matrix in with respect to matrix multiplication?
- How does the vector space relate to the concept of linear transformations on ?
Tip:
Remember that each element of can be represented as a linear combination of four basis matrices, which simplifies many vector space computations.
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Vector Spaces
Linear Algebra
Formulas
Matrix addition formula: A + B = \begin{pmatrix} a + e & b + f \\ c + g & d + h \end{pmatrix}
Scalar multiplication of a matrix: \alpha A = \begin{pmatrix} \alpha a & \alpha b \\ \alpha c & \alpha d \end{pmatrix}
Theorems
Vector Space Axioms
Suitable Grade Level
Undergraduate
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