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Definition
A subspace of R^n is any set H in R^n that has three properties:
(1) The zero vector is in H
(2) For each u and v in H, the sum u + v is in H
(3) For each u in H and each scalar c, the vector cu is in H. |
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Definition
| The rank of a matrix A, denoted by rankA, is the dimension of the column space of A. (the rank A is the number of pivot columns of A) |
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Definition
| An eigenvalue of A is a scalar lambda (&), such that the equation Ax = &x for some scalar & |
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Definition
| An egienvector of an n x n matrix A is a nonzero vector x such that Ax = &x for some scalar &. |
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Definition
| The set of all solutions of Ax = &x where & is an eigenvalue of A. Consists of the zero vector and all eigenvectors corresponding to &. |
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Definition
| A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix, that is, A=PDP^(-1) for some invertible matrix P and some diagonal matrix D. |
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Definition
| If A and B are n x n matrices in R^n, then A is similar to B if there is an invertible matrix P such that P^(-1)AP = B or equivalently, A = PBP^(-1) |
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Definition
| Two vectors in R^n u and v are orthogonal if u dot v = 0. |
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Definition
| A set S of vectors such that u dot v = 0 for each distinct pair u, v in S. |
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Definition
| An orthogonal basis for a subspace W in R^n is a basis for W that is also an orthogonal set. |
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Definition
| The set W^(perp) of all vectors orthogonal to W (Read as W perpendicular or W perp). |
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| Orthogonal Projection onto a Subspace |
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Definition
| The unique vector yHAT in W such that y-yHAT is orthogonal to W. |
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| Orthogonal Projection onto a Vector |
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Definition
| the vector yHAT defined by yHAT = ((y dot u)/(u dot u))u |
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