Part 11 : Row Space, Column Space, and Null Space

Row Space

The span of row vectors of any matrix, represented as a vector space is called row space of that matrix.

or

If we represent individual columns of a row as a vector, then the vector space formed by set of linear combination of all those vectors will be called row space of that matrix.

Assuming a 3x3 matrix A

Dividing matrix A into row vectors we get

Row space of matrix A will be

All the linear combinations of row vectors : a1, a2 and a3

Column Space

Similar to row space, column space is a vector space formed by set of linear combination of all column vectors of the matrix.

Column vectors of matrix A

Column space of matrix A will be

All the linear combinations of column vectors : a1, a2 and a3

Both of these spaces have same dimension (same number of independent vectors) and that dimension is equal to rank of matrix. Why?

Because, rank of matrix is maximum number of linearly independent vectors in rows or columns and dimension is maximum number of linearly independent vectors in a vector space (like column space or row space).

Rows and columns of a matrix have same rank so the have same dimension.

Null Space

We are familiar with matrix representation of system of linear equations.

Here A is coefficient matrix, X is variable matrix and 0 represents a vector of zeros

We can also find it’s solution (values of variables for which the equation above is satisfied) using Gaussian Elimination algorithm.

If we take a set of all possible solution vectors (all possible values of “x”), then the vector space formed out of that set will be called null space.

Or

Null space contains all possible solutions of a given system of linear equations.

Taking an example

Solution vector of system of linear equations above is

So this system of linear equations has two vectors in null space.

Null space contains all the linear combinations of solution and zero vector

Null space always contains zero vector.

Red line represents the null space of system of linear equations

Nullity

Dimension of null space is called nullity.

Nullity of the system above is 1.

Read Part 12 : Vectors

You can view the complete series here
I’m now publishing at avni.sh

Connect with me on LinkedIn.