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Linear Algebra
Definitions
20
Mathematics
Undergraduate 3
03/05/2009

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Term
Define Ax
Definition
The linear combination of the columns of A using the corresponding entries in x as weights: x1a1 + x2a2 + ... + xnan
Term
Define Span{v} & its geometric interpretation
Definition
The set of all scalar multiples of v. Geometrically, it is visualized as a set of points on a line in R^n
Term
Define Span{u,v} & its geometric interpretation
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
Define Span{v1,...,vp}
Definition
If v1,...,vp are in R^n, then the set of all linear combinations of v1,...,vp is denoted by Span {v1,...,vp} and is called the subset of R^n 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
Define Linear Independence
Definition
An indexed set of vectors {v1,...,vp} in R^n is said to be linearly independent if the vector equation:
x1v1 + x2v2 + . . . + xpvp = 0
has only the trivial solution.
Term
Define Linear Dependence
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
Define 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
Define a Standard matrix of a linear transformation
Definition
The matrix A such that T(x) = Ax in the domain of T. A = [T(e1)...T(en)]
Term
Define 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
Define One to one transformation
Definition
A mapping T:R^n->R^m such that each b in R^m is the image of at most one x in R^n.
Term
Define Onto transformation
Definition
A mapping T:Rn->Rm such that each b in R^m is the image of at least one x in R^n.
Term
Define 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.
Term
When are {v1,v2} linearly dependent
Definition
If one of the vectors is a multiple of the other.
Term
Describe AB using the columns of AB
Definition
Each column of AB is a linear combination of the columns of A using weights from the correspoind column of B.
Term
For AB, what does row-i(AB)= ?
Definition
row-i(AB)=row-i(A)B
Term
What does the transpose of a product of matrices equal?
Definition
The transpose of a product of matrices equals the product of their transposes in the reverse order.
Term
Define a subspace of R^n
Definition
A subspace of R^n is any set H in R^n that has three properties:
a. The zero vector is in H
b. For each u and v in H, the sum u+v in H
c. For each u in H and each scalar c, the vector cu is in H
Term
Define the column space of a matrix
Definition
The column space of a matrix A is the set Col A of all linear combinations of the columns of A
Term
Define the null space of a matrix
Definition
The null space of a matrix A is the set Nul A of all solutions to the homogeneous equation Ax = 0.
Term
Define a basis for a subspace
Definition
A basis for a subspace H of R^n is a linearly independent set in H that spans H
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