Lecture 7 Ax= 0, Nullspace
- Elimination of A. Rank = Number of Pivot in the Matrix.
2. Pivot column, and Free column. Free columns don’t create a new dimension. Free columns can be represented by pivot columns. So in x2 and x4 can be assigned with any number.
3. For each free column corresponding x, let the x be 0 or 1 to generate special solutions. And the solution space is the linear combination of these special solutions.
Number of Columns — Number of pivot = Number of Free columns.
Solution =a * Free Column 1 + b * Free Column 2 + ……
4. rref: Reduced Row Echelon Form: That is the simplest form.
Now the original equation is simplified as:
5. Whole Process:
Ax=0 -> Ux=0 ->Rx=0
6. Notice the F matrix and red numbers, I matrix, Blue numbers.
7. Find a special solution: assume the x for free columns to 1 and 0.
f1 = 1, f2 = 0, f3 = 0 : solution 1
f1 = 0, f2 = 1, f3 = 0: solution 2
f1 = 0, f2 = 0, f3 = 1: solution 3
8. Clever solution: Because I*-F + F*I = 0
Lecture 8 Ax=b
- Augment Matrix.
2. The last column shows the conditions on solvable, especially the last row.
3. Solvability condition on b:
Ax = b is solvable when b is in C(A)<easy to understand to think about vector space.>
If a combination of rows of A gives zero rows, the same combination of entires of B must give 0. <e.g: last row.> It is a special case of the above condition.
4. Particular solution.
Step 1: A->U.
Step 2: Solvability check.
Step3: Assume b under condition(Last row).
Step 4: Replace bs in the matrix.
Step 5. Assume free column x to be 0
Step6. Get new equations. (free columns disappeared, b-rows become target)
Step 7. solve the equation.
5. General Solutions.
General Solutions = Particular solutions + Nullspace.
Because: Nullspace, adding a zero doesn’t change the b(Otherwise, b will change.).
One guy + subspace. No subspace for Ax=b particular, due to assumed b.
6.
Originally, the Nullspace will be a subspace go through the origin. However, the particular solution shifts the subspace away from the origin. So it is no longer a sub-space.
7. Assume matrix A is a m x n matrix.
Full column rank: n = r, m ≤ n in this case. Otherwise pivot < n
It means there will be no free column. the nullspace will only be Ax=0, x = 0, the origin. There might be no solution at all or a unique solution.
If there exist a particular solution: Ax = b: solution = the particular solution(a point) + origin . <0, 1 solution>
Full row rank: m = r, m ≥n in this case. Otherwise pivot < m. n-r = n-m free columns.
It means there will be some free columns. There will be a solution subspace for Nullspace. Solution exists.
Full rank: m = n = r: No free column, R = I, only one unique solution other than origin. Invertible matrix.
8. Summarizing:
The rank tells you everything about the number of solutions.
when r<m and r<n, there will be some free columns. However, there may not exist a solution for Ax=b.
