exam 1 Flashcards
(32 cards)
rank
number of pivots in RREF
free variables
their columns have no pivots
linear equation
equation with polynomials of degree 1
coefficient matrix
matrix of coefficients of linear system
augmented matrix
matrix of coefficients and solutions of linear system
what three operations can be preformed on a matrix
1) scale
2) swap
3) replace with itself plus scalar multiple of another matrix
leading entries
the first nonzero number in a row of a matrix
pivots
leading entries that are 0
row echelon form
1) any rows of all zeros are at the bottom of the matrix
2) first leading entry of each row are to the right of all leading entries above (diagonal)
reduced row echelon form
1) it’s in row echelon form
2) all leading entries are pivots
3) every pivot is the only nonzero entry in its column
forward phase of RREF algorithm
1) use row operations to get 1 in top left entry
2) use row operations to clear out entries below the pivot
3) repeat for all rows
backward phase of RREF
1) start at bottom right pivot and use row operations to clear all entries above
2) repeat
solution set
collection of a1… an in R^n that satisfies the equation
row equivalence
two matrices are row equivalent if you can transform one into the other with row operations (A ~ B)
every matrix is row equivalent to exactly one…
reduced row echelon matrix
If A~B then the systems have the same…
solution set
a system is inconsistent…
when the RREF form has a row [0…0 1] 0 does not equal 1.
homogeneous linear system
linear system with solution constants that are all equal to 0
a1x + a2x = 0
if a system has free variables, then it has…
infinitely many solutions
trivial solution
all variables equal to 0
if A is an m by n matrix then rank(A) <=
min (m,n)
ex: if A is 3 by 4 then possible ranks are 0,1,2,3
number of free variables equals
n - r (cols minus rank)
if n - r is 0 then the solution is unique
polynomial interpolation
there exists a unique line through only two distinct points on a plane
vandermande theorum
given n + 1 points in the plane with distinct x values, there is a unique polynomial a0 + a1x + anx^n
