Finite-Dimensional Linear Algebra Thumbnails Lists [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∈Fm×n is diagonal if Aij=0 for all i≠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 u1,u2,…,un in vector space V, if v∈sp{u1,u2,…un}, thensp{u1,u2,…un,v}=sp{u1,u2,…un}This leads to the concept of a basis, a spanning set containing the fewest possible elements. Definition of a BasisLet u1,u2,…un be vectors in a vector space V. We say that {u1,u2,…un} 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 u1,u2,…uk be vectors in V, and let α1,α2,…αk be scalars in F. Then the linear combination of the vectors with scalars we call weights is: α1u1+α2u2+⋯+αkukAny 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∈V⇒u+v∈Vscalar multiplication: α∈F,v∈V⇒αv∈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 α,β∈F called addition and multiplication.addition: α+β∈Fmultiplication: αβ∈F F is a field iff (if and only if) the operations satisfy the following properties: A. Axioms for AdditionAssociative property of additionfor all α,β,γ∈F, (\alpha + \.. Show More Previous 1 Next
