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Definition
| Rectangular array of numbers |
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Definition
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Term
| Elementary Row Operations |
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Definition
1. Multiply a row by constant
2. Add a multiple of a row to another and replace row
3. Swap or interchange rows |
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Definition
1. All non zero rows are above zero rows
2. Each leading non zero entry of a row is in a column to the right of the one above it
3. All entries in a column beneath a leading column must be 0 |
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Definition
1. Each leading entry is equal to 1
2. Each lead entry is the only non zero entry in its column |
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Definition
| A variable x in a system of linear equations in n unknowns x1 . . . xn is free if there are no restrictions placed on that variable |
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Definition
| A linear system is consistant <==> the rightmost column of the augmented matrix is NOT a pivot column |
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Definition
V is an element of the span {v1,...vn} if v is a linear combination of v1,...,vn
Span {v1...vn} is the set of all linear combinations of v1...vn |
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Definition
| Ax=b where A is a matrix and x,b are vectors |
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Definition
A set of vectors {v1...vn} is said to be linearly dependent if there exists scalars x1...xn, not all of them zero, such that x1v1 + . . .+ xnvn = 0 (ld relation)
So at least one of the vectors can be written as a lc of the others |
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Definition
| A set {v1...vn} are linearly independent if x1v1 + xnvn= 0 has only the trivial solution x1 = xn = -0 |
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| Tests for linear dependence |
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Definition
1. Set contains 0 vector
2. v1 and v2 arent 0 vector and v1 = kv2 (scalar multiples)
3. A set contains more vectors than entries |
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Definition
| A transformation T from Rn to Rm is a rule that assigns to each vector x in Rn a vector T(x) in Rm |
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Term
Domain of T
Codomain of T
Range of T |
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Definition
Rn domain
Rm range
Set of all images t(x) |
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| A transformation is linear if: |
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Definition
1. T(u + v) = T(u) + T(v) for all u,v in the domain of T
2. T(cu) = cT(u) for all u and scalars c |
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Term
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Definition
T(0) = 0 vector
T(cu + dv) = cT(u) + dT(v) |
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