First Sentence:
In many practical problems, relations among variables may be simply expressed directly or approximately in terms of linear functions so that it is necessary to investigate such functions.
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Key Phrases - Statistically Improbable Phrases (SIPs):
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pairwise orthogonal nonzero vectors, algebraic cofactor, pairwise orthogonal unit vectors, linear space over complex numbers, given solution vector, zero linear transformation, independent solution vectors, identity linear transformation, adjacent interchanges, pairwise orthogonal vectors, elementary divisors, inverse linear transformation, using elementary operations, three row vectors, real quadratic forms, elementary column operations, largest independent set, nonsingular linear transformation, using elementary row operations, nonzero minors, algebraic complement, real diagonal matrix
Key Phrases - Capitalized Phrases (CAPs):
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Linear Algebra, Matrix Operations, Inner Product Spaces, Jordan Canonical Form of Matrices, Find the Jordan, Matrices Similar
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