Term
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Definition
zero rows below nonzero rows
lead term is to the right of lead term in row above |
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Term
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Definition
| First nonzero term in a row, is a pivot if lead term is not in the same column as any other lead term |
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Term
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Definition
| Any variable multiplied by a column that doesn't contain a pivot |
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Term
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Definition
| Variable that is multiplied by a column that has a pivot in it, any variable that isn't a free variable |
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Term
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Definition
| All pivots are 1's, zero rows below nonzero rows, no lead terms that aren't pivots, all lead terms are to the right of the lead term in the row above them. |
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Term
| Consistent vs. Inconsistent |
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Definition
Consistent systems have one or more solutions
Inconsistent systems do not have a solution (in row echelon form an inconsisten augmented matrix would have a row of zeroes with the last column nonzero) |
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Term
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Definition
| A solution with no free variables, pivots are in every column |
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Term
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Definition
| a 3rd vector is in the span of the first 2 vectors if the augmented matrix of the three vectors has a solution |
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Term
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Definition
| For all intents and purposes in this class a vector is one of the columns. |
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Term
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Definition
Ax=0
where A is a matrix, always has it least 1 solution by setting x to the zero vector |
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Term
| Independent System vs Dependent System |
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Definition
matrix A is independent if Ax=0 only has one solution (the trivial x=0 case, meaning that no column is a linear combination of any other vector(s) )
matrix A is dependent if Ax=0 has multiple solutions |
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Term
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Definition
| A matrix of all 0's but with the top left to bottom right diagonal filled with 1's (must be square) |
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Term
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Definition
Invertible matrix A must be square and have and follow the rule:
A*A-1=Identity Matrix |
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Term
| Vectors e1, e2, e3 etc... |
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Definition
| These vectors are column vectors that are all 0 except with a 1 in the subscript position. When multiplied by a matrix, only that subscript column is perserved, every other column is destroyed. Useful for figuring out transformations as TA has column 1 of T(e1) etc... |
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Term
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Definition
Linear Transformations are represented as T and follow these rules:
T(0)=0
T(x+y)=T(x)+T(y)
cT(x)=T(cx) |
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Term
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Definition
| A Linear Transformation from Rn to Rm is onto if the matrix A associated with T spans Rm meaning A must have m pivots, or a pivot in each row. m must be lesser than or equal to n |
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Term
One-to-One ness
(or injective-ness) |
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Definition
| A Linear Transformation from Rn to Rm is one to one if the matrix A associated with T is linearly independent meaning A must have n pivots, or a pivot in each column. m must be greater than or equal to n |
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Term
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Definition
Aij=Aji
note that diagonals will stay the same
AT+BT=(A+B)T
(AB)T=BTAT |
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