Chapter 2:
Theorem 2.4: A system of linear equations with augmented matrix [A l
b ] is consistent if and only if b is a linear combo of the columns of A.
X[column 1] + y[column 2] = [b]
Definition: If S={V ,V , … V } is a set of vectors in R , then the set of
1 2 k
linear combos of S is called the Span(S). If Span(S)= R thennS is
n
called a spanning set for R .
Example 2.19: Show that R = Span ( [ 1 2 ], [ 2 4]). Show that an
arbitrary vector [ a b] can be written as a combo of the two vectors
like x[V1] + y[V 2 = [ a b]
Definition: A set of Vectors V 1V 2….V ik linearly dependent if there are
scalars c1,c2,…c kt least one of which is not zero such that C V +1C1V 2 2
+ …CkVk= 0
n
Theorem 2.5: Vectors V ,V 1….2 in k are linearly dependent if and
only if at least one of the vectors can be expressed as a linear
combination of the others
Example 2.22: Any set of vectors containing the zero vector is linearly
dependent. For if 0,V ,…V are in R , then we can find a nontrivial
2 m
combination of the form C 0 1 C V +2…C2V = 0 my metting C =1 and 1
C2…C =m
Theorem 2.6: Let V ,V1,…2V be mcolumn) vectors in Rn and let A be
the nxm matrix [V1,V2,..Vm] with these vectors as its columns. Then
V1,V 2….V mre linearly dependent if and only if the homogeneous
linear system with augmented matrix [A l 0] has a nontrivial solution.
Proof: cV 1cV 2….cV =m0 means that [ C ,C …C1] 2s amsolution of the
system [V ,1 ,2.V l m]
Theorem 2.7: The rows of a matrix will be linearly depende

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