Math 233A
Introduction to Linear Algebra
Chapter 3: Vector Spaces and Subspaces
Section 3.2: The Nullspace of A: Solving Ax = 0
3.2) The Nullspace of A: Solving Ax = 0
Today’s Lecture
Math 233A
Today’s Lecture
Nullspace
Echelon Matrices
Solutions to Ax = 0
Nullspace
The subspace which contains all solutions to Ax =0.
Echelon Matrices
Similar to upper triangular matrices, echelon matrices are
the result of elimination but this time its applied to
rectangular matrices.
Solutions to Ax = 0
The solutions to Ax = 0 form an important part of the
general solution to Ax = b.
Review
A brief review of the key ideas of this lecture.
Prepared by: Frederick H. Willeboordse
Review
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3.2) The Nullspace of A: Solving Ax = 0
Nullspace
Math 233A
Today’s Lecture
Nullspace
Echelon Matrices
Where we are: We have seen in section 3.1 that the matrix equation Ax = b is
only solvable if b is in the column space C(A).
Solutions to Ax = 0
Review
We now continue our systematic investigation into the solutions of Ax =b. It
will turn out that the solutions can be split into two parts, one part of which is
generated by solving Ax = 0.
Can have only x = 0 as solution. Trivial solution.
Can have infinitely many solutions.
Example:
The solutions form the plane
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3.2) The Nullspace of A: Solving Ax = 0
Math 233A
Today’s Lecture
Nullspace
Nullspace
Echelon Matrices
Solutions to Ax = 0
The solutions to the m by n matrix equation Ax = 0 form a
subspace in Rn called the nullspace of A . It is denoted by N(A).
Review
Example:
A is a 1 by 3 matrix
The solutions form the plane
N(A) is a plane that is a subspace of R3
Note that the plane
Prepared by: Frederick H. Willeboordse
is not a subspace as it doesn’t go through 0.
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3.2) The Nullspace of A: Solving Ax = 0
Nullspace
Math 233A
Today’s Lecture
Nullspace
Echelon Matrices
A good way to express the nullspace is by considering some special solutions
and then to equate the nullspace to all the linear combinations of those special
solutions.
Solutions to Ax = 0
Review
Example:
Choose some simple z and y
and calculate x.
Some good choices are (y,z) = (1,0) , and (y,z) = (0,1).
Special solutions:
,
How many would there be?
Prepared by: Frederick H. Willeboordse
5
3.2) The Nullspace of A: Solving Ax = 0
Nullspace
Math 233A
Today’s Lecture
Nullspace
Echelon Matrices
Now there are two things to note:
Solutions to Ax = 0
Review
Firstly: There are free variables, i.e. variables that we can choose freely. In the
example on the previous slide we have 1 equation and 3 variables. This implies
that 2 can be chosen while the 3rd is fixed. In general, if there are more variables
than equations (i.e. if n > m), there will be free variables.
Secondly: The method used to find some special solutions on the previous slide
doesn’t work very well for larger coefficient matrices A. For example if A is a 4
by 7 matrix, how do you know how many free variables there are? Therefore,
we need a more systematic way to identify suitable free and non-free variables.
Prepared by: Frederick H. Willeboordse
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Math 233A
3.2) The Nullspace of A: Solving Ax = 0
Echelon Matrices
Today’s Lecture
Nullspace
Echelon Matrices
Solutions to Ax = 0
What is a good way to find the special solutions?
Review
As before, we use elimination. The only difference is that now all matrices are
allowed even when they have more columns than rows.
Consequently, not all rows will have pivots. The more general upper triangular
matrix then obtained from elimination is called echelon matrix.
Pivot variables: x1, x2, x6
Free variables: x3, x4, x5, x7
N(U) is a subspace of R7
Echelon Matrix
Pivot columns = columns with pivots
What we see is that it is very convenient to take the pivot variables as the ones to
be calculated and the remaining ones as free since this will always work. It should
be stressed though that this is a choice and that in principle any variable can be
free as long as the total number of free variables is correct.
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3.2) The Nullspace of A: Solving Ax = 0
Echelon Matrices
Math 233A
Today’s Lecture
Nullspace
Echelon Matrices
Systematic method to find the special solutions:
Solutions to Ax = 0
Review
1.
2.
3.
4.
Find the free variables
Set one free variable to 1 and the rest to zero
Solve Ux = 0 for the pivot variables
Continue steps 1-3 until each free variable has been set to 1 once
Once we have the echelon matrix, we can continue with elimination just as we
did for the Gauss-Jordan method by first eliminating the entries above the pivots
and then by dividing the rows by their pivots (if they have one). This yields the
so-called reduced row echelon matrix which makes back substitution even easier.
Prepared by: Frederick H. Willeboordse
Reduced Row Echelon Matrix
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3.2) The Nullspace of A: Solving Ax = 0
Solutions to Ax = 0
Math 233A
Today’s Lecture
Nullspace
Echelon Matrices
Quick summary for finding the solutions of Ax = 0. These solutions form the
nullspace N(A).
Solutions to Ax = 0
Review
N(A) consists of all the linear combinations of the special solutions.
There is one special solution for every free variable.
Free variable
Free variable
Pivot variable
Example:
Note that if n > m, then A has one or more columns without pivots and
consequently free variables giving special solutions. In other words, every
free column (i.e. a column without a pivot) leads to one special solution.
Prepared by: Frederick H. Willeboordse
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3.2) The Nullspace of A: Solving Ax = 0
Review
Math 233A
Today’s Lecture
Nullspace
Echelon Matrices
The nullspace N(A) of the matrix A is formed by all the solutions to Ax = 0.
Solutions to Ax = 0
Review
The nullspace can be found from all the linear combinations of the special
solutions to Ax = 0.
The special solutions are obtained by setting one of the free variables to one,
the other free variables to zero and solving the system repeatedly until all free
variables have been one once.
This implies that every free column leads to a special solution.
The free variables are found from the (reduced row) echelon matrix that is
obtained when applying elimination to A.
Prepared by: Frederick H. Willeboordse
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