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Mathematics

[Finite-Dimensional Linear Algebra] 4.1 + 4.2 + 4.3 The Determinant Function Analysis of matrices is central in the study of finite-dimensional linear algebra, because any linear operator mapping one finite-dimensional space into another can be represented by a matrix. The simplest kind of matrix ia a diagonal matrix. The matrix `A \in F^{m \times n}` is diagonal if `A_{ij} = 0` for all `i \neq j` To simplify matrix-based calculations, we want to transform matrices into .. Show More
[Finite-Dimensional Linear Algebra] 2.5 + 2.6 + 2.7 Basis and Dimension Lemma For a set of vectors ` u_1, u_2,\dots , u_n` in vector space `V`, if `v \in sp \{ u_1, u_2,\dots u_n \} `, then`` sp \{ u_1, u_2,\dots u_n, v \} = sp \{ u_1, u_2,\dots u_n \} ``This leads to the concept of a basis, a spanning set containing the fewest possible elements. Definition of a BasisLet `u_1, u_2,\dots u_n` be vectors in a vector space `V`. We say that `\{ u_1, u_2,\dots u_n \}` is.. Show More
[Finite-Dimensional Linear Algebra] 2.4 Linear Combinations and Spanning Sets Definition of a Linear CombinationLet `V` be a vector space over a field `F`, let `u_1, u_2,\dots u_k` be vectors in `V`, and let `\alpha _1, \alpha _2,\dots \alpha _k` be scalars in `F`. Then the linear combination of the vectors with scalars we call weights is: ``\alpha _1 u_1+\alpha _2u_2+\cdots + \alpha _ku_k``Any linear combination of vectors in subspace `S` belongs to `S`, as subspaces are.. Show More
[Finite-Dimensional Linear Algebra] 2.2 Vector Spaces + 2.3 Subspaces The elements of vector space `V` are called vectors, and the elements of the corresponding field `F` are called scalars. Definition of a Vector SpaceLet `F` be a field and let `V` be a nonempty set with two operations defined with respect to these sets: (vector) addition: `u, v \in V \Rightarrow  u + v \in V`scalar multiplication: `\alpha \in F, v \in V \Rightarrow \alpha v \in V` `V` is a vecto.. Show More
[Finite-Dimensional Linear Algebra] 2.1 Fields Definition of a fieldLet `F` be a nonempty set with two defined operations for `\alpha , \beta \in F` called addition and multiplication.addition: `\alpha + \beta \in F`multiplication: `\alpha\beta \in F` `F` is a field iff (if and only if) the operations satisfy the following properties: A. Axioms for AdditionAssociative property of additionfor all `\alpha , \beta , \gamma \in F`, ``(\alpha + \.. Show More

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