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| Echelon Form - Requirements |
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Definition
All 0 rows are at the bottom.
Leading entry is to right of leading coeff in row above
All entries below a "pivot" are 0 |
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| REF + #'s above are nonzero + pivots are 1 |
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| Pivot variables and free variables |
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| Pivots - variables that are pivots (x1,x2,etc) in the RREF matrix. Free variables - every other variable |
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| A linear system is consistent if ___ |
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Does not have a row like [0,0,0 | b] (b is a non zero number).
0x+0y+0z≠b |
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Can combine scalars with vectors.
c1A+c2B+c3C = D
to get another vector |
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| Span(v) is the set of all vectors of the form c*v |
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| Span(v1,v2) is the collection of all vectors in the direction of v1 or v2 |
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| Span(u,v) for plane vs line situation |
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| Is v a multiple of u? Yes - span line. No? Span plane. |
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| Determine if w is in Span(u,v) (u,v,w are vectors)? |
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Definition
| Check if linear system corresponding to u,v|w is consistent (write as augmented form with vectors as columns) |
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| m x n matrix (columns, rows) |
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| The matrix is equal to it's transpose |
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| Solve system with LU decomp |
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Definition
If Ax=b, then
Lc=b -> Ux=c |
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Definition
set of objects I can linear combine
(ex: matricies, polynomials) |
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Definition
It's a subspace of V if:
H is a subset of V if:
It shares a 0 vector with V
Sums of things in H are in H
cU is in H if U is in H |
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Definition
1) Find U using rowops
2) Take the opposite sign of row ops and insert correctly. |
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