Find a Basis for Null(C).

Solution

Step 1) Compute rref(C).

Step 2) Write the equivalent system.

Step 3) Write the general solution using scalars for the free variables.

If

and

the general solution is given by

Step 4) Write in vector form:

The Null Space is spanned by :

The nullity of C is:

Find the rank of C.

We are ready for an important result.

Lecture 10 Notes

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d) Write out AT . What is Rank(AT)?

c) How does the Rank(A) compare to the Dimension of Col(A)?

b) Find a basis for Col(A).

a) Let U = rref(A). Are the columns of U linearly independent?

Is Col(U) = Col(A)? Explain.

Given

Definition Column Space : If A is an mxn matrix, each column is an m-tuple of real numbers and can be considered as a vector in Rm . The span of these n vectors forms a subspace of Rm called the column space of A, written Col(A) .

III. The Column Space of a Matrix

Verify the Rank and Nullity Theorem for:

The Rank and Nullity Theorem

If A is an mxn matrix

Rank(A) + Nullity(A) = n ,

the number of columns of A.

Rank(C) + Nullity(C) = ?

What is

Compute the reduced row echelon form U of A. Let rref(A) = U

Are the row vectors of U linearly independent?

Is Row(U) = Row(A)? Explain.

Given

(Hint: Think of each row as a single vector. A contains 5 row vectors)

What is Dim(Row(A))? (What is the Dimension of the row space of A?)

etc.

Let

Find a basis for Row(A):

Definition Row Space: If A is an mxn matrix, each row is an n-tuple of real numbers and can be considered as a vector in R1xn . The span of these m vectors forms a subspace of R1xn called the row space of A, written Row(A) .

I. The Row Space and Rank of a Matrix

Goals: Know the definitions of Row Space, Column Space, Rank and nullity of a matrix. Know the Rank and Nullity Theorem, The Matrix Dimension Theorem.

Row Space, Column Space and Null Space

What is the nullity of C?

Definition Nullity: The Null Space of an mxn matrix A is the set of all n-vectors such

that Ax = 0. Null(A) or N(A) = { x e Rn | Ax = 0 }. The Nullity of A is the dimension of Null(A).

II. The Rank and Nullity Theorem

Change some of the elements of C. What do you observe about Rank(C) and Rank(CT)?

Mathcad has the command word rank() .

Find Rank(C) and Rank(CT).

Is this true or false? Explain.

Two row equivalent matrices have the same row space.

Determine Rank(B), given:

Definition Rank: The Rank of a matrix A is the dimension of the row space of A:

Rank(A) = Dim(Row(A))