MATH1012 - Lecture 8

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MATH1012 - Lecture 7
Skeleton Slides

Row and Column Rank

• Row/column/null rank is the dimension of the row/column/null space
  • The dimension of a matrix is equal to the # of non-zero rows/columns in row echelon form
• Row rank is equal to column rank

Rank-Nullity Theorem

If $A$ is an $m\times n$ matrix, then: $rank(A)+nullity(A)=n$
i.e. ‘rank + nullity = number of columns’

Every column is either basic or non-basic:

• Basic $-$ contributes to rank
  • These are columns with leading entries in row echelon form
• Non-basic $-$ contributes to nullity

Explanation of Sowiso Assignment 2 (Q5) in lecture recording

Invertible Matrices

• A matrix is invertible if:
  • $A$ has full rank (i.e. rank = #rows)
  • The rows and columns of $A$ are linearly independent

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Annotated Slides
MATH1012 - Lecture 9