Linear Algebra E3
Quiz yourself by thinking what should be in
each of the black spaces below before clicking
on it to display the answer.
Help!
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| What is a Linear Transformations? | show 🗑
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| show | 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? | show 🗑
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| show | Fro each vector v in V, the vector w=T(v) is called the image of v under T
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| show | 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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| show | 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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| show | 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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| show | 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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| show | 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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| show | 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 | show 🗑
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| Define length (norm) | show 🗑
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| Define the distance between u and v | show 🗑
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| show | u has a length = 1; thus ||u|| = 1
u = 1/||u|| * v
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| show | 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 | show 🗑
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| show | |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 | show 🗑
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| Inner Product Axioms | show 🗑
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| Inner Product Space | show 🗑
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| show | 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 | show 🗑
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