Module: Bases and dimension

Learning Outcomes

Bases and Dimension

• Basis = Maximal linearly independent set
• Natural coordinate basis for \({\mathbb R}^n\): columns of \(I_n\)
• Dimension of a linear space = Size of the basis
• Identify basis for fundamental spaces of \(A\) using (pivot) rows/columns of \(A\), as well as non-zero rows/cols of rref(\(A\)) and rref(\(A^T\))
• Dimension of all fundamental spaces of \(A\) in terms of rank and shape of \(A\)
• If \(A\) is \(m\times n\) has rank \(k\), column and row space of \(A\) have dimension \(k\)
• If \(A\) is \(m\times n\) has rank \(k\), null space \(A\) has dimension \(n-k\), null space of \(A^T\) has dimension \(m-k\)

Readings

Experiential Learning