Linear Algebra E3
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| What is a Linear Transformations? | Let V and W be vector spaces. the mapping T:V --> W is called a LINEAR TRANSFORMATION if and only if:
T(cu+v) = cT(u)+T(v)
for every choice of u and v in V and scalars c in R.
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| When is T a linear operator? | When V=W the T is a linear operator
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| What is the difference between an image and the range of a linear transformation? | The image is of a vector.
The range is the "map" (or collection of images from the transformation)
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| Define Image | Fro each vector v in V, the vector w=T(v) is called the image of v under T
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| Define Range | The range of an operator, T, denoted R(T), is the collection of all images of the vectors v in V
R(T)={T(u)|v E V}
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| Define Null Space | Let V and W be vector spaces, for a linear transformation T:V-->W, the NULL SPACE of T, denoted N(T), is defined as:
N(T)={v E V|T(v)=0}
"for every vector v in V the linear transformation of V maps v to the zero vector"
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| Null Space and Range Theorem | 1. The null space of T is a subspace of V
2. The range of T is a subspace of W
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| Eigenvalue and Eigenvector | Let A be an n x n matrix. A number \\ is called an EIGENVALUE of A provided that there exists a non zero vector in n space such that
Av = \\v
Every non zero vector satisfying this equation is called an eignevector of A corresponding to the eigenvalue \
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| Eigenvalue - Eigenvector Pair | We say \\ and v form an eigenvalue - eigenvector pair: A will have infinitely many eigenvectors associated with \\ such that:
A(cv)=c(Av)=c(\\v)=\\(cV)
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| Eigenspaces | An eigenspace is the set of all eigenvectors corresponding to an eigenvalue along with the zero vector.
V(\\)={v E n space | Av = \\v}
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| Define Dot Product | Let u and v be vectors in euclidean n-space, the dot product of u and v is given by
u*v = u1v1 + u2v2 + u3v3 +...+ unvn
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| Define length (norm) | ||v|| = sqrt(v1^2 + v2^2 +...)
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| Define the distance between u and v | ||u-v|| = sqrt((u-v)*(u-v))
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| Define unit vector | u has a length = 1; thus ||u|| = 1
u = 1/||u|| * v
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| Define angle between u and v | the cosine of the angle theta between the vectors v and w, is give by
cos(theta) = (v*w)/(||v|| ||w||)
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| Define Orthogonal | The vectors u and v are orthogonal if the angle between them is pi/2, cos(theta) = 0; thus
(u*v)/(||u|| ||v||) = 0 and
u*v = 0
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| Cauchy Schwartz Inequality | |x*y| <= ||x|| ||y||
ie. the absolute value of two vectors is less than or equal to the product of the magnitude of 2 vectors
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| Define inner product space | Let V be a vector space over all real numbers. An inner product on V is a function that associates with each pair of vectors in u and v in V a real number, denoted <u,v>, that satisfies all four inner product axioms
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| Inner Product Axioms | 1. <u,u> => 0 and <u,u> = 0 if and only if u=0 (the zero vector)
2. <u,v> = <v,u> (symmetry)
3. <u+v,w> = <u,w> + <v,w> (linear property)
4. <cu,v> = c<u,v>
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| Inner Product Space | A vector space V with an inner product
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| Orthogonal Sets | We say that u and v are orthogonal provided that <u,v> = 0. The set V={v1,v2,v3,...vn} is orthogonal if the vectors are mutually orthogonal to each other
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| Define Mutually Orthogonal | every vector in a set is orthogonal to each other
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