27.2: The Four Fundamental Spaces

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In the lecture on Change Basis, we talked about four subspaces based on a matrix \(A\):

Row space of \(A\): linear combination of all rows of \(A\)

Column space of \(A\): linear combination of all columns of \(A\)

Null space or kernel of \(A\): all \(x\) such that \(Ax=0\)

Null space of \(A^{\top}\): all \(y\) such that \(A^{\top}y=0\)

In this course we represent a system of linear equations as \(Ax=b\). The matrix \(A\) can be viewed as taking a point \(x\) in the input space and projecting that point to \(b\) in the output space.

It turns out, everything we need to know about \(A\) is represented by four fundamental vector spaces. Two of the four spaces are easily defined as follows:

Row space of \(A\): linear combination of all rows of \(A\)

Column space of \(A\): linear combination of all columns of \(A\)

The other two fundamental spaces are defined by a concept called the Null Space. The Null space is calculated by finding all the solutions to the homogeneous system \(Ax=0\). The final two fundamental spaces are defined as follows:

Null space or kernel of \(A\): all \(x\) such that \(Ax=0\)

Null space of \(A^{\top}\): all \(y\) such that \(A^{\top}y=0\)