Week 9- Kernel and Image Bases, Matrix Multiplication

Null Space

For a linear mapping L : V → W, the set of all input vectors that are mapped to 0W is called the kernel of L, or the null space of L.

What is special about the columns of A, a standard matrix?

Let L:Rn→Rm be a linear transformation, and A its standard matrix. Then the columns of A are a generating set for Im(L)

What is the algorithm for finding a basis of Im(L)

Let A be the standard matrix for L.

• Put A into RREF.

• Look at the columns of A with leading ones. The corresponding columns in the original A are a basis for Im(L).

What is the algorithm for finding a basis of ker(L)

Let A be the standard matrix for L.

• Put A into RREF.

• Add an extra column of 0’s to get an augmented matrix.

• Write down the general solution to the corresponding system in vector form

• The vectors associated with the free variables are a basis for the kernel

How to find dimension from basis?

number of vectors in basis = dimension

How to interpret dim(Im(L)) from RREF

# of columns with leading ones

How to interpret dim(Ker(L)) from RREF

# of columns without leading ones

Rank-Nullity Theorem

If L:Rn→Rm is a linear transformation, then n=Dim(Im(L))+Dim(Ker(L))

When is L injective?

iff Ker(L) = {0} (0 dimensional)

When is L surjective?

• leading 1’s in RREF in every column and row

• matrix must be square (input dim = output dim)

Combining matrices

If A is an m×n matrix, and B is an p×m matrix, the we define the product BA to be the p×n matrix BA=Bc1 | Bc2 | Bc3 | ... | Bcn where the ci are the columns of A