Week 1 Flashcards
What is the linear combination of vectors?
A vector v is a linear combination of vectors v1,…,vk iff there are scalars c1,c2,…,ck such that
v=c1v1+c2v2+…+ckvk
What is a linear equation?
A linear equation in the n variables x1,x2,…,xn is an equation that can be written in the form
a1x1+a2x2+…+anxn=b
A solution of this linear equation is a vector [s1,s2,…,sn] so that
a1s1+a2s2+…+ansn=b
A system of linear equations is a finite set of linear equations and a solution to a system of linear equations is a vector that is simultaneously a solution to all equations in the system.
What is row echelon form?
A matrix is in row echelon form iff
1) any all zero rows are at the bottom
2) in each non zero row, the first non zero entry (the leading entry) is to the left of any leading entries below it
What is reduced echelon form?
A matrix is in reduced row echelon form iff
1) it is in row echelon form
2) the leading entry in each non zero row is 1
3) each column containing a leading 1 has zeros everywhere else
What is a homogeneous system of linear equations?
A system of linear equations is homogeneous iff the constant term in each equation is zero
What is theorem 2.4?
A system of linear equations with augmented matrix [A\b] is consistent iff b is a linear combination of the columns of A.
What is linear independence?
A set {v1,v2,…,vk} of vectors in Rn is linearly independent iff the only solution to the equation
c1v1+c2v2+…+ckvk=0
Is c1=c2=…=ck=0. A set of vectors is linearly dependent if it is not linearly independent.
What is theorem 2.5?
Vectors v1,v2,…,vk in Rn are linearly dependent iff at least one of them can be expressed as a linear combination of the others.
What is lemma (linear dependence)?
Two vectors v1 & v2 are linearly dependent iff they are scalar multiples of each other.
What is theorem 2.6?
Let v1,…,van be (column) vectors in Rn & let
A=[v1,v2,…,vm]
Be the nxm matrix with columns v1,…,vm. Then v1,…,vk are linearly dependent iff the homogenous linear system with augmented matrix [A\0] has a non trivial solution.
What is theorem 2.8?
If m>n then any set of m vectors in Rn is linearly dependent.
