Subspaces Flashcards
Column space is subspace of Rᵐ or Rⁿ ?
Column space is subspace of Rᵐ
How to find basis of column space
Pivot columns of A
Dimension of column space of A
Dim(col(A)) = Rank of A
Null space is subspace of Rᵐ or Rⁿ ?
Null space is subspace of Rⁿ
How to find basis of null space
AX=0
1.) In echelon form, identify free and pivot variables.
2.) Express pivot variables in terms of free variables.
3.) Form general solution in terms of free variables
4.) Set each free variable to 1 (one at a time) and all others to a form the individual basis vectors.
Another name of null space
Kernel
Nullity formula
Nullity = n - rank(A)
here n: number of columns
If AX=0 has only one solution then what about its null space
If AX=0 has only one solution i.e. trivial solution then the null space of the matrix is zero vector.
Row space is subspace of Rᵐ or Rⁿ ?
Row space is subspace of Rⁿ
How to find basis of row space
Non zero rows in the reduced echelon form
Left null space is subspace of
Left null subspace of subspace of Rᵐ
Equation of left null space
YᵀA=0
Dimension of left null space
m - rank(A)
here m:rows of A
or m: columns of Aᵀ
Summary
Explain orthogonality of subspaces
