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Review Linear Algebra (9/11/21) #3

Closed ahy69195 closed 2 years ago

ahy69195 commented 3 years ago

Review linear algebra

ahy69195 commented 2 years ago

Let A be an m x n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false. a. For each b in R^m, the equation A x = b has a solution. b. Each b in R^m is a linear combination of the columns of A. c. The columns of A span R^m. d. A has a pivot position in every row.

Why does A have to have a pivot position in every row?

ahy69195 commented 2 years ago

Wait nvm, I figured it out. This only applies if the Ax = b fulfills all b in the space R^m, so that is why A has to have a pivot position in every row; if there wasn't a pivot position in every row, the "columns of A could not span R^m" (every b in R^m is a linear combination of the columns of A). This is also just a coefficient matrix, not augmented*.

Micky774 commented 2 years ago

Question for you: Suppose that we have a matrix A that satisfies statement c). Does that imply that the inverse of A exists? Why or why not? In particular, when does the inverse of A exist. Try not to just google this, and frame the response wrt to the statements a-d that are listed above.