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6. From Method 1 of solving the least-squares problem, the LS solution is unique if and only if the columns of A are lin
Posted: Mon May 09, 2022 10:35 am
by answerhappygod
- 6 From Method 1 Of Solving The Least Squares Problem The Ls Solution Is Unique If And Only If The Columns Of A Are Lin 1 (140.7 KiB) Viewed 23 times
6. From Method 1 of solving the least-squares problem, the LS solution is unique if and only if the columns of A are linearly independent. By Method 2, the LS solution is unique if and only if ATA is invertible. Prove that these two methods give an equivalent condition for uniqueness by filling in the missing parts of the proof below. Proof that the columns of A are linearly independent if and only if ATA is invertible: Assume the columns of A are linearly independent. Consider the equation, (ATA)x= 0. a. (4 pts) Prove that if the vector x is in Nul(ATA), then x is in Nul(A). Hint: What do we know about Nul(AT)? b. (3 pts) Use (a) to explain why ATA is invertible. Hint: Use the assumption that the columns of A are linearly independent. Therefore, if the columns of A are linearly independent, then ATA is invertible. Next, assume ATA is invertible. Consider the equation, Ax= 0. c. (4 pts) Prove if Ax 0, then x=0. Therefore, if ATA is invertible, then the columns of A are linearly independent.