Math 221 Flashcards
echelon form
- all nonzero rows are above any rows of zeros
- leading entry of row is to the right of leading entry in row above
- entries in column below leading entry is zero
reduced echelon form
- leading entry in nonzero row is 1
2. each leading 1 (pivot) is only nonzero entry
Theorem 1
each matrix is row equivalent to one and only one reduced echelon matrix
Theorem 2
linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column with form [0…0 b] when b does not equal 0
Theorem 3
If A is mxn matrix with columns a1…an and if b is in Rm, the matrix equation Ax=b has the same solution set as the vector equation which has the same solution set as the system of the linear equation
This Theorem serves to show different ways to view linear equations
Theorem 4
A is an m x n matrix and the following statements are logically equivalent:
1. for each b in Rm, equation Ax=b has a solution
2. each b in Rm is a linear combination of columns of A
3. columns of A span Rm
4. A has pivot position in every row
this is for a coefficient matrix only
Theorem 5
If A is an m x n matrix, u and v are vectors in Rn and c is a scalar then
A(u+v) = Au +Av
A(cu) = c(Au)
Theorem 6
Suppose equation Ax=b is consistent for given b and let p be a solution. Then solution set of Ax=b is set of all vectors of form w = p +Vh where Vh is any solution of homogenous equation Ax=0
linear independence
an indexed set of vectors is linearly independent if it has only the trivial solution
Theorem 7
an indexed set S of 2 or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others
Theorem 8
If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. Any set in Rn is linearly dependent if vectors > entries
Theorem 9
If a set S in Rn contains zero vector then the set is linearly dependent
