Chapter 2 Flashcards
Def: consistent/inconsistent
If a system has at least one solution, it is said to be consistent
Otherwise, it is said to be inconsistent
Def: equivalent
Two systems of equations with the same solution set are said to be equivalent
Def: coefficient and augmented matrix
The augmented matrix is the one that has the | solutions part
Coeff: [A]
Augment: [A|b]
Def: elementary row operations
1) multiply row by non scalar
2) add a multiple of one row or another
3) swapping two rows
Def: row equivalent
Two matrices are said to be row equivalent if there exists a sequence of elementary row operations that transform A into B
A~B
Def: reduced row echelon form
RREF
1) zero rows on the bottom
2) first nonzero entry in each row is 1
3) leading one in each non zero row is to right of previous
4) leading one is only nonzero entry in column
Theorem 2.2.2
If A is a matrix, then A has a unique RREF R
Gauss-Jordan Elimination
Basically put matrix into RREF
Def: free variable
Let R be the RREF of a coefficient matrix of a system of linear equations
If the column of R does not contain a leading one, then we call xj a free variable
Def: homogenous system
A system of linear equations is said to be a homogeneous system if the right-hand side only contains zeros
[A|0]
Theorem 2.2.3
The solution set of a homogeneous system of m linear equations in n variables is a subspace of R^n
Def: solution space
The solution set of a homogeneous system is called the solution space of the system
Def: rank
The rank of a matrix is the # of leading ones Demoted rank(A)
Theorem 2.2.4
For any mxn matrix A we have rank(A)
Theorem 2.2.5 (system rank theorem)
Let A be the coefficient matrix of a system of m linear equations in n unknowns [A|b]
1) rank(A) < rank([A|b]) iff the system has no solution
2) if [A|b] has a solution, then the system has n-rank(A) free variables
3) rank(A)=m iff [A|b] has a solution for every bER^m
Theorem 2.2.6 (solution theorem)
Let [A|b] be the augmented matrix, [C|d] be the matrix in RREF If rank(A) = rank(C) = k LESS THAN n, the so,union set of the augmented matrix is either an empty set or an n-k dim plane in Rn
Theorem 2.2.7 (theorem from class)
1) {v1,…,vk}CR^m is linearly independent iff the rank of the mxk matrix [v1…vk] is k
2) {v1,…,vm}CR^m is linearly independent iff Span{v1,…,vm}=R^m
Another theorem~
If A is an mxn matrix, then rank(A) less or equal m AND less or equal to n
