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18.03 midterm 2
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Terms in this set (18)
underdamping of harmonic oscillator mx"+bx'+kx=0
b^2<4mk (b is small relative to m and k)
overdamping of harmonic oscillator mx"+bx'+kx=0
b^2>4mk (b is large relative to 4 m and k)
critical damping of harmonic oscillator mx"+bx'+kx=0
b^2=4mk
leading entry of a row
leftmost nonzero entry in a nonzero row.
rules of echelon form
(1) All nonzero rows are above any rows of all zeros.
(2) Each leading entry is in a column to the right of the leading entry of the row above it.
(3) All entries in a column below the leading entry are all zeros
rules of rref
(4) The leading entry in each nonzero row is 1
(5) Each leading 1 is the only nonzero entry in its column.
pivot column
a column that contains a leading 1 in rref OR a column that contains the first nonzero entry in a nonzero row in ref
pivot variable
variable whose corresponding column is a pivot column
free variable
variables that don't correspond to pivot columns; can be set to any value
Nullspace of a matrix
the solution set of the homogeneous equation A x = 0.
dimension of nullspace
the number of free variables
nullity of matrix
dimension of nullspace of matrix
rank of matrix
number of nonzero rows in echelon form
rank nullity theorem
rank A + nullity A = the number of columns of A
span of vector space
set of all linear combos of vector space
span(v)=cv
Column Space of a Matrix
the span (set of all possible linear combinations) of its column vectors (columns of original matrix)
basis of column space of matrix
the columns of the ORIGINAL matrix that contain a pivot
basis of nullspace of vector
the columns (excluding one corresponding to constants) you get when you solve Ax=b
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