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Term
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Definition
| An equation that can be written in the form a1X1 + a2X2 + ... + anXn = b |
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Term
| System of Linear Equations |
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Definition
A collection of one or more linear equations involving the same variables. A system of linear equations has either:
1) no solution, or
2) Exactly one solution, or
3) Infinitly many solutions. |
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Term
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Definition
| If a system of linear equations has either one solution or infinitly many solutions |
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Term
| Elementary row operations |
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Definition
1) (replacement) Replace one row by the sum of itself and a multiple of another
2) (Interchange) Interchange two rows
3) (Scaling) Multiply all entries in a row by a nonzero constant. |
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Term
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Definition
| If a system of linear equations has no solution |
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Term
| The two fundamental questions about a linear system |
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Definition
1) Is the system consistent (does at least one solution exist)?
2) If a solution does exist, is it unique? |
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Term
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Definition
1) All nonzero rows are above any rows of all zeros
2) each leading entry of a row is in a column to the right of the leading entry of the row above it.
3) all entries in a column below a leading entry are zeros.
4) The leading entry in each nonzero row is 1.
5) Each leading 1 is the only nonzero entry in its column. |
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Term
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Definition
| A location in matrix A that corresponds to a leading 1 in the reduced echelon form of A. A Pivot column is a column of A that contains a pivot position. |
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Term
| Algebraic Properties of R ^ n |
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Definition
For all vectors U, V, and W in R ^ n and all scalars C and D:
1) U + V = V + U
2) (U + V) + W = U + (V + W)
3) U + 0 = 0 + U = U
4) U + (-U) = -U + U = 0
5) C(U + V) = CU + CV
6) (C + D)U = CU + DU
7) C(DU) = (CD)U
8) 1U = U |
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Term
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Definition
| A sum of scalar multiples of vectors. The scalars are called the weights. |
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