- #1
Maxwhale
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Homework Statement
Show that Ax = b has a solution if and only if b is in CS(A).
Homework Equations
The Attempt at a Solution
Ax = b
b [tex]\in[/tex] CS(A) means
d1A1 + d2A2+ ...+ dnAn = b
and I am lost
For a solution to exist for Ax=b, it means that there is a set of values for the variable x that satisfies the equation and makes it true. In other words, when we substitute these values into the equation, it will result in the given value of b.
CS (A) stands for the column space of the matrix A. It represents the span of all possible linear combinations of the columns in A. In the context of the equation Ax=b, it means that the value of b must be in the column space of A for a solution to exist.
We can determine if b is in CS (A) by checking if the columns of A are linearly independent. If they are linearly independent, then b is in the column space of A. We can also use methods such as row reduction or calculating the rank of A to determine if b is in CS (A).
No, a solution cannot exist for Ax=b if b is not in CS (A). This is because the equation represents a system of linear equations, and the column space of A represents all possible solutions to the system. If b is not in CS (A), it means that there is no combination of the columns of A that can produce the value of b.
Yes, there are special cases where a solution may still exist if b is not in CS (A). This can happen when the matrix A is not square, and there are more than one solution to the system of linear equations. In this case, b may not be in CS (A) but a solution to Ax=b still exists.