Let A be a 4x3 matrix and letbandcbe two vectors inR^4. We are told that the system Ax=bhas a unique solution. What can you say about the number of solutions of the system Ax=c?

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- Jan 31st 2011, 06:25 AMvcf323Solutions of Linear Systems: Matrix Algebra
Let A be a 4x3 matrix and let

**b**and**c**be two vectors in**R**^4. We are told that the system A**x**=**b**has a unique solution. What can you say about the number of solutions of the system A**x**=**c**? - Jan 31st 2011, 06:48 AMHallsofIvy
A "4 by 3" matrix maps a 3 vector into a 4 vector. More specifically, it maps all of $\displaystyle R^3$ to a subspace of $\displaystyle R^4$, of dimension 3 or

**less**. If the dimension were less than 3 then A would have non-trivial kernel and more than one vector would be mapped to a specific vector in that subspace. Saying that Ax= b has a solution means that b is in that subspace. Saying that Ax= b has a**unique**solution means that a unique vector is mapped into every member of that subspace. So there are**two**possibilities for the number of solutions of Ax= c depending upon whether c happens to be in that subspace or not.