Term

Definition
zero rows below nonzero rows
lead term is to the right of lead term in row above 


Term

Definition
First nonzero term in a row, is a pivot if lead term is not in the same column as any other lead term 


Term

Definition
Any variable multiplied by a column that doesn't contain a pivot 


Term

Definition
Variable that is multiplied by a column that has a pivot in it, any variable that isn't a free variable 


Term

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. 


Term
Consistent vs. Inconsistent 

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) 


Term

Definition
A solution with no free variables, pivots are in every column 


Term

Definition
a 3rd vector is in the span of the first 2 vectors if the augmented matrix of the three vectors has a solution 


Term

Definition
For all intents and purposes in this class a vector is one of the columns. 


Term

Definition
Ax=0
where A is a matrix, always has it least 1 solution by setting x to the zero vector 


Term
Independent System vs Dependent System 

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 


Term

Definition
A matrix of all 0's but with the top left to bottom right diagonal filled with 1's (must be square) 


Term

Definition
Invertible matrix A must be square and have and follow the rule: A*A^{1}=Identity Matrix 


Term
Vectors e_{1}, e_{2}, e_{3} etc... 

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 T_{A} has column 1 of T(e_{1}) etc... 


Term

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) 


Term

Definition
A Linear Transformation from R^{n} to R^{m} is onto if the matrix A associated with T spans R^{m} meaning A must have m pivots, or a pivot in each row. m must be lesser than or equal to n 


Term
OnetoOne ness
(or injectiveness) 

Definition
A Linear Transformation from R^{n} to R^{m} 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 


Term

Definition
A_{ij}=A_{ji}
note that diagonals will stay the same
A^{T}+B^{T}=(A+B)^{T}
(AB)^{T}=B^{T}A^{T} 

