Download Unit Three Review

Unit Three Review
Linear Spaces
Review of terms:
 A Linear Space: is a nonempty set that is closed under addition and multiplication
by a scalar (real or complex) that satisfies under addition: commutative property,
associative property, additive identity, and additive inverse property and under
multiplication by a scalar (k1  k 2 )v1  k1v1  k 2 v1 , (k1k 2 )v1  k1 (k 2 v1 ) ,
k1 (v1  v2 )  k1v1  k1v2 and 1* v  v .
 Subspace: a nonempty subset S of a linear space V that inherits the same
properties as the space itself. In other words, a subspace must be closed under
addition and under multiplication by a scalar.
 Linear Combination: A vector w is a linear combination of a set of
vectors v1 , v2 ,..., vn if there exist scalars c1 , c2 ,..., cn such that
w  c1v1  c2 v2  ...  cn vn .
 Span of a set L(S): is the set of all linear combination c1v1  c2 v2  ...  cn vn of
the elements v1 , v2 ,..., vn of the set S.
 Linear Dependent Set: set of vectors v1 , v2 ,..., vn such that there exists at least one
nonzero scalar for which c1v1  c2 v2  ...  cn vn  0 .

Linear Independent Set: set of vectors v1 , v2 ,..., vn for which if
c1v1  c2 v2  ...  cn vn  0 , then c1  c2  c3  ...  cn  0 .




Basis of a Linear Space: is a set of linearly independent set that spans the space.
Row Space of a Matrix: is the subspace spanned by the rows of the matrix.
Column Space of a Matrix: is the subspace spanned by the columns of the matrix.
Rank of a Matrix: is the maximum number of linear independent rows in the
reduced row echelon form of the matrix.
Review of some facts:
 The linear span of a set S is a subspace.
 The system Ax  b is consistent if and only if rank (A | b) =rank (A).
 A basis of a linear space is not unique but the number of elements is the same for
all bases.
 A vector in a linear space can be expressed uniquely in terms of the basis
elements.
Review Questions:
1. Give an example of a linear space and state the reasons for your choice.
2. Describe the subspaces of R 3 .
3. For what values of a, b, and c is the vector ( a, b, c ) a linear combination of the
vectors: (1,2,4), (1,1,2) . Write one vector that can be written asa linear
combination.
4. For what values of a, b, and c is the vector ( a, b, c ) a linear combination of the
vectors: (1,2,4), (3,1,2) and (5,1,3) .
5. If an nxn matrix A has n independent rows, describe the solution set of the
system Ax  b .
6. If an nxn matrix A has r (r < n) independent rows, describe the solution set of the
system Ax  b .
7. For what values of x:
2 1  1
3 1 2 


5 2 x 
does the matrix have rank 2?
8. Given the matrix A::
2 1 1 
1 3 4 


5 5 6
(a) Are the rows of A linearly dependent or independent? Explain
(b) Describe the row space and the column space of the matrix
9. Find a basis for the row space of the matrix A:
 2 1  1
1 3 4


  1 4 2 