MATH307 April 2006
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Question 01 (a)
Let PA = LU be
(a) Find a basis for each of the four fundamental subspaces.
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This matrix A in this question is given in the form of an LU decomposition with partial pivoting where P is just the matrix which swaps the rows of A. The matrix P also would not affect R(A), N(A), R(AT) where we need to look at the matrix U (echelon form) to find the bases.
To find a basis for the R(A), we must identify the pivot columns in matrix U, which turns out to be the first and second column.
The pivots are highlighted in red.
The basis for R(A) is then the corresponding columns in A to the pivot columns in U.
The pivot columns of A are highlighted in red.
To find N(A), we can set and solve for x. This would give us 2 equations and 2 free variables.
The basis of R(AT) can be taken from the rows of U that contain pivots.
To find the basis for N(AT), we must row reduce AT and solve for x in
From row reducing AT, you should acquire the matrix
Solving for x, you end up with 2 equations and 1 free variable.