What is Linear Algebra?Date: 09/06/97 at 07:26:23 From: wchoy Subject: College level I was wondering if you could help me with College level maths. I attend a University in Australia and I have absolutely no idea about linear algebra. To illustrate this, the lecturer has been talking about kernels and dimensions and from what I gather, a kernel is associated with corn and a dimension is on of three that we exist in. ANY sort of help would be greatly appreciated! Warren Choy Date: 09/12/97 at 12:04:42 From: Doctor Rob Subject: Re: College level Linear algebra is the study of vector spaces and of linear maps between them. A vector space over a field F is a set of elements V and two operations. The first operation is "plus," and it maps any pair of elements in V to an element of V. V with the operation "plus" is an abelian group. The second operation is "times," and it maps a pair consisting of an element of F and an element of V to an element of V. "times" is associative, distributes over "plus" in V and + in F, and has the left identity the element 1 in F. An example is V = the set of all n-tuples of real numbers, F = the field of real numbers. "plus" is coordinate-wise addition, i.e., ((x1,...,xn),(y1,...,yn)) |--> (x1+y1,...,xn+yn). "times" is defined by scalar multiplication: [a,(x1,...,xn)] |--> (a*x1,...,a*xn). Every vector space has a "basis," a set of elements {v(i)} such that (1) they are linearly independent, i.e., if Sum x(i)*v(i) = 0, and each x(i) is in F, then each x(i) = 0; and (2) every element in V can be written in the form Sum x(i)*v(i), where every x(i) is in F. It turns out that every basis has the same size. The number of elements in any basis is called the dimension of the vector space. A linear map from one vector space V to another W is a function T: V --> W such that for any a and b in F and any x and y in V, T(a*x+b*y) = a*T(x)+b*T(y). If you know T(v(i)) for any basis, you know what T does to any element of V. The kernel of a linear map T is the set of all v in V such that T(v) = 0. It turns out that this is a subset of V which is a vector space using the same "plus" and "times" as are defined for V. Such things are called vector subspaces. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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