Linear Algebra

... real numbers, reals greater than 0, n-tuples of reals natural numbers: {0, 1, 2, . . . }, complex numbers interval (open, closed) of reals between a and b sequence; like a set but order matters vector spaces vectors, zero vector, zero vector of V bases, basis vectors standard basis for Rn matrix rep ...

... real numbers, reals greater than 0, n-tuples of reals natural numbers: {0, 1, 2, . . . }, complex numbers interval (open, closed) of reals between a and b sequence; like a set but order matters vector spaces vectors, zero vector, zero vector of V bases, basis vectors standard basis for Rn matrix rep ...

Abstract Vector Spaces, Linear Transformations, and Their

... Corollary 1.8 If V is a vector space and S, T ⊆ V , then the following hold: (1) S ⊆ T ⊆ V =⇒ span(S) ⊆ span(T ) (2) S ⊆ T ⊆ V and span(S) = V =⇒ span(T ) = V (3) span(S ∪ T ) = span(S) + span(T ) (4) span(S ∩ T ) ⊆ span(S) ∩ span(T ) Proof: (1) and (2) are immediate, so we only need to prove 3 and ...

... Corollary 1.8 If V is a vector space and S, T ⊆ V , then the following hold: (1) S ⊆ T ⊆ V =⇒ span(S) ⊆ span(T ) (2) S ⊆ T ⊆ V and span(S) = V =⇒ span(T ) = V (3) span(S ∪ T ) = span(S) + span(T ) (4) span(S ∩ T ) ⊆ span(S) ∩ span(T ) Proof: (1) and (2) are immediate, so we only need to prove 3 and ...

Trace of Positive Integer Power of Real 2 × 2 Matrices

... The computation of the trace of matrix powers has received much attention. In [5], an algorithm for computing Tr Ak , k ∈ Z is proposed, when A is a lower Hessenberg matrix with a unit codiagonal. In [6], a symbolic calculation of the trace of powers of tridiagonal matrices is presented. Let A be a ...

... The computation of the trace of matrix powers has received much attention. In [5], an algorithm for computing Tr Ak , k ∈ Z is proposed, when A is a lower Hessenberg matrix with a unit codiagonal. In [6], a symbolic calculation of the trace of powers of tridiagonal matrices is presented. Let A be a ...