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33 Cards in this Set
- Front
- Back
Linear Equation
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a1x1 + a2x2 + ... + anxn = b
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System of Linear Equations
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Collection of 1 or more linear equations
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Linear System
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Collection of 1 or more linear equations
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Solution
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A list (s1, s2, ..., sn) of numbers that make an equation a true statement when (s1, s2, ..., sn) are substituted for (x1, x2, ..., xn)
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Solution Set
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all possible solutions
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Two linear systems are equivalent when
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they share the same solution set
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Systems of linear equations can have _____ solution(s)
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no solution
1 solution ∞ solutions |
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Consistent Linear System
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either:
1 solution ∞ solutions |
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Inconsistent Linear System
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no solution
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Augmented matrices of two linear systems are row equivalent. They have the same _____
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Solution Set
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Two Fundamental Questions
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Does at least 1 solution exist?
If a solution exists, is it only one? |
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Unique Solution
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Only one solution
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Row Equivalence
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elementary row operations transforms one matrix into the other
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Leading Entry
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the leftmost nonzero entry of a row
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Matrix in Echelon Form has
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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 |
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Additional Conditions for Reduced Row Echelon Form
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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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Uniqueness of the Reduced Echelon Form
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Each matrix is row equivalent to one and only one reduced echelon matrix.
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Pivot Position
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A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A.
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Pivot Column
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A pivot column is a column of A that contains a pivot position.
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Partial Pivoting
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Reduction of round-off errors by using largest absolute value when choosing the pivot for a pivot column
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Basic Variables
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variables corresponding to pivot columns
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Free Variable
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Not Basic Variables
Each different choice of the free variable determines a (different) solution of the system, and every solution of the system is determined by a choice of the free variable. |
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zero vector
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Bolded 0
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Weights for:
y = c1v1 + ... + cpvp are |
c1, ..., p1
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The vector field for:
y = c1v1 + ... + cpvp is |
y
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True of False:
The presence of a free variable in a system does not guarantee that the system is consistent. |
It could be shown the more likely there is a free variable the more likely a system will be inconsistent.
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Linear Dependence Relation
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c1v1 + c2v2 + ... + cpvp = 0 is called a linear dependence relation among v1, ... , vp when the weights are not all zero.
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v1, ..., vp are Linearly Dependent, we mean ...
(pg 56) |
{v1, ..., vp} is a linearly dependent set
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v1, ..., vp are Linearly Independent, we mean ...
(pg 56) |
{v1, ..., vp} is a linearly independent set
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Linear Dependence
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Non-Trivial Solution
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Trivial Solution
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c1=c2=c3=0
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Non-Trivial Solution
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c1=c2=c3≠0
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Linearly Independent iff
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Ax = 0 has only the trivial solution
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