Understanding Least Squares and Column Space

What does it mean if there is no solution to the equation Ax = b?

• b is not in the column space of A

• A is not an n-by-k matrix

• b is in the column space of A

• x is not a vector in Rk

What is the goal of finding an approximate solution x-star?

• To find a solution in the null space of A

• To maximize the distance between Ax and b

• To minimize the distance between Ax and b

• To make Ax equal to b

What is the least squares solution?

• The vector that maximizes the error

• The exact solution to Ax = b

• The vector that is orthogonal to b

• The closest vector in the column space to b

What is the orthogonal complement of the column space?

• The column space of A transpose

• The row space of A

• The null space of A transpose

• The null space of A

How can we find the least squares solution using A transpose?

• By solving Ax = b directly

• By solving A transpose times Ax = A transpose b

• By finding the inverse of A

• By maximizing the error between Ax and b