Shared Flashcard Set

Details

Linear Algebra
Definitions
12
Mathematics
Undergraduate 3
03/05/2009

Additional Mathematics Flashcards

 


 

Cards

Term
Ax
Definition
The linear combination of the columns of A using the corresponding entries in x as weights

a1x1 + a2x2 + … + anxn
Term
Span {v}
Definition
The set of all scalar multiples of v. Visualized as the set of points on the line in R3 and through v and 0. Geometric interpretation in R3 is a line through the origin.
Term
Span {u,v}
Definition
If u and v are nonzero vectors in R3, with v not a multiple of u, the Span {u,v} is the plane in R3 that contains u, v and 0. In particular, Span {u,v} contains the line in R3 through v and 0. Geometric interpretation in R3 is a plane through the origin.
Term
Span {v1 ... vp}
Definition
If v1, . . . ,vp are in Rn, then the set of all linear combinations of v1, . . . ,vp is denoted by Span {v1 . . . vp} and is called the subset of Rn spanned (or generated) by v1, . . . ,vp. That is, Span {v1 . . . vp} is the collection of all vectors that can be written in the form
c1v1 + c2v2 + . . . + cpvp
with c1, . . . , cp scalars.
Term
Linearly Independent
Definition
An indexed set of vectors {v1 . . . vp} in Rn is said to be linearly independent if the vector equation:
x1v1 + x2v2 + . . . + xpvp = 0
has only the trivial solution.
Term
Linearly dependent
Definition
The set of vectors {v1 . . . vp} is said to be linearly dependent if there exists weights c1, . . . , cp, not all zero, such that:
c1v1 + c2v2 + . . . + cpvp = 0.
Term
Linear Transformation
Definition
A transformation (or mapping) T is linear if:
(i) T(u + v) = T(u) + T(v) for all u, v in the domain of T
(ii) T(cu) = cT(u) for all u and scalars c
Term
Standard matrix (for a linear transformation T)
Definition
The matrix A such that T(x) = Ax in the domain of T.
Term
Matrix product (AB)
Definition
If A is an m x n matrix, and if B is an n x p matrix with columns b1, . . . , bp, then the product AB is the m x p matrix whose columns are Ab1, . . . , Abp. That is
AB = A[b1 b2 … bp] = [Ab1 Ab2 … Abp]
Term
One-to-One (mapping)
Definition
A mapping T: Rn Rm such that each b in Rm is the image of at most one x in Rn.
Term
Onto (mapping)
Definition
A mapping T: Rn Rm such that each b in Rm is the image of at least one x in Rn.
Term
LU Factorization
Definition
The representation of a matrix A in the form A = LU where L is a square lower triangular matrix with ones on the diagonal (a unit lower triangular matrix) and U is an echelon form of A.
Supporting users have an ad free experience!