Introduction
Recall that a setS spans a vector spaceV if every vector inV can be written as a linear combination of vectors inS. A set S is abasis for a vector space V if 1)S spansV and 2)S is linearly independent.
Thedimension of a vector spaceV, dim(V), is the number of vectors in a basis. If a basis for a vector space, V, consists of only the 0 vector then dim(V) = 0. If the number of basis vectors for a vector space is finite we call the vector spacefinite dimensional, otherwise we call the vector spaceinfinite dimensional.
Note that a basis for a vector space is not unique; however two different bases for the same vector space will contain the same number of vectors. In addition, if two vector spaces have the same basis then they are the same vector space.
Exercises: Let V =R3.
a. Give an example of a set of vectors inR3 that spansV but is not linearly independent.
b. Give an example of a set of vectors inR3that is linearly independent but that does not spanV.
c. One basis forR3 isS ={(1,0,0),(0,1,0),(0,0,1)}, give another example of a basis forR3.
d. What is the dimension ofRn? Nullspace
In Lab 11, we found that the general solution,solution set, of the homogeneous systemAx= 0 whereA=
1 2 2 4
, is a subspace ofR2. The solution set to Ax= 0 is called theNullspaceofA. To find a basis for the nullspace of a matrix
Type:null(The Name of the Matrix,‘r’)
The dimension of the nullspace of a matrix A is called the nullity of A.
Note that if MATLAB reports that the nullspace is empty then the only vector in the nullspace is the 0 vector.
Vector Spaces 53 Exercises: DefineA=
1 2 2 4
, B=
1 2 3 4 5 6 0 1 2
, and
M =
1 0 5 0
0 1 3 −1
−2 0 1 4
.
a. Find a basis for the nullspace ofA and the nullity ofA.
b. Find a basis for the nullspace ofB and the nullity ofB.
c. From parts a and b make a conjecture about the nullspace and nullity of invertible matrices.
d. Compare the nullity of Awith the nullity of AT and the nullity ofB with the nullity ofBT.
e. Note that matrix M is not square but we can still find a basis for the nullspace ofM, so find a basis for the nullspace of M and the nullity of M.
Rowspace and Columnspace
The rowspace of A is the set of vectors that can be written as linear com- binations of the rows of A. The dimension of the rowspace ofA is the rank ofA, rank(A). Similarly the columnspace of Ais the set of vectors that can be written as linear combinations of the columns of A. The dimension of the columnspace ofAis also therank(A).
The vectors of one basis for the rowspace ofAare the nonzero rows ofAin reduced row echelon form. Unlike the rows ofA, the columns ofAare affected by row operations. So to find the vectors of a basis for the columnspace of A, you can putAin reduced row echelon form and then identify the columns with the leading ones. The corresponding columns in the originalAwill create a basis for the columnspace ofA.
Exercises: Using matricesA, B, andM from above a. Find a basis for the rowspace of Aandrank(A).
b. Determinerank(A) +nullity(A).
c. Determine if the basis for the rowspace ofA from part a spansR2. d. If an n×n matrix is invertible make a conjecture about the relationship
between the rowspace of the matrix andRn.
e. Would a similar result to that in d hold for the columnspace of an n×n invertible matrix? If you are unsure try finding the columnspace of A.
54 Exploring Linear Algebra Labs and Projects with MATLABR e. Find a basis for the rowspace and columnspace of M.
f. Determinerank(M) +nullity(M).
g. Make a conjecture about the sum of the rank and nullity of a square matrix.
What is the sum of the rank and nullity of a matrix in general?
Theorems and Problems
For each of these statements, either prove that the statement is true or find a counter example that shows it is false.
Theorem 45. A is invertible if and only if the nullspace of A = ~0 and nullity(A) = 0.
Theorem 46. Ann×nmatrixAis invertible if and only if the rowspace of A=Rnand rank(A) =n.
Problem 47. IfAis anm×nmatrix thenrank(A) +nullity(A) =m.
Problem 48.rank(A) =rank(AT).
Now how many different statements can you think of that are equivalent to saying that a square matrix,A, is invertible?
IfAis ann×nmatrix the following are equivalent statements:
1. Ais invertible.
2. |A| 6= 0.
3. The reduced row echelon form ofAisIn.
4. Acan be written as a product of elementary matrices.
5. The systemAx=bhas exactly one solution for alln×1 vectorsb.
6. The systemAx= 0 has only the trivial solution.
7. The nullspace ofA=~0 andnullity(A) = 0.
8. The rowspace ofA=Rn, the columnspace ofA=Rn, andrank(A) =n.
Vector Spaces 55