A ? consists of a closed set V of vectors, a field F of scalars and satisfies all operations of vector addition and scalar multiplication
A subset U of some vector space V such that U has these properties: (1) the zero vector of V is in U; (2) U is closed under vector addition; and (3) U is closed under multiplication by scalars
Let V be a vector space. We say that a subset of V is ? if whenever λ's such that λv's=0 then λ's=0.
The ? of S is the set of all linear combinations of S
A subset is a ? if it is linearly independent and spans the vector space
The ? of V is the unique number of vectors in any basis of V.
A sequence of vectors where the set of vectors are a basis of V
1. All zero rows occur at the bottom of the matrix2. in every non zero row the first non-zero entry is a 1 (pivot)3. each pivot occurs to the right of all pivots in the rows above.
Reduced Echelon Form
The first non-zero entry in any row is the only non-zero entry in its column
If r is a root of p(λ) then the number of times (λ-r) is a factor of p(λ) is the AM of r
Using (A-λI)x=0, the ? is the set of eigenvectors corresponding to λ
the set of all solutions of Ax=λ(x) where λ is an eigenvalue, consists of all eigenvectors and the zero vector
Dimension of eigenspace
If there exists an eigenvalue λ such that the GM<AM then the matrix is said to be ?
When two matrices A, B and another (invertible) matrix S satisfy A=S⁻¹BS
A matrix is ? iff A has a set of n linearly independent eigenvectors
A real (n x n) matrix Q such that Q^(-1)=Q^T or Q^T Q=I