Lay Chapter 4,
Vector Spaces

dim(domain) == dim(kernel) + dim(range)

Consider a certain 3x4 matrix.

In[1]:=

a = ({{1, 2, 3, 4}, {0, 1, 2, 3}, {0, 0, 1, 2}}) ;

The matrix a can be interpreted as the matrix of a linear map,
and the dimensions of certain vector spaces associated with that linear map will satisfy a well-known relationship.

In[2]:=

(* dim (domain)  dim (kernel) + dim (range) *)Last[Dimensions[a]] Length[NullSpace[a]] + MatrixRank[a]

Out[2]=

True

Let's write some procedures which will make that relationship more transparent.

In[3]:=

dimDomain[a_] := Last[Dimensions[a]]

dimKernel[a_] := Length[NullSpace[a]]

dimRange[a_] := MatrixRank[a]

Test those procedures on the matrix a.

In[6]:=

dimDomain[a]

dimKernel[a]

dimRange[a] 

dimDomain[a] == dimKernel[a] + dimRange[a]

Out[6]=

4

Out[7]=

1

Out[8]=

3

Out[9]=

True

Basis for the Kernel of a Linear Map

Consider a certain 2x4 matrix.

In[20]:=

a = ({{1, 2, 3, 4}, {0, 1, 2, 3}}) ;

The matrix a can be interpreted as the matrix of a linear map,
We seek a basis for the null space of this linear map.

In[22]:=

NullSpace[a] ;

%//MatrixForm

Out[23]//MatrixForm=

( {{2, -3, 0, 1}, {1, -2, 1, 0}} )


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