Question

hi,guys, is there any one read the answer for the first exam? It says because Ax=[0,1,0] has no solution, the null space is only the zero vector. Could anybody explain a little bit about this?

Answer 1

Could you write the entire question here, please? I couldn't find the stuff you mentioned in your question in the exam solution.

Answer 2

Hi,guys, i think when there is no solution to a particular linear system Ax=b means b is not in the column space of A. ~ I don't understand your explanation very well~oh~

Answer 3

The question says that (1,1,1) has no solution and that (0,1,0) has exactly one solution. So (0 1 0) is in the column space but (1 1 1 ) is not.

Answer 4

Here is a link to Exam 1. http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/ax-b-and-the-four-subspaces/exam-1/MIT18_06SCF11_ex1s.pdf

**1 (24 pts.) This question is about an m b n matrix A for which \[Ax = \left[\begin{matrix}1 \\ 1\\ 1\end{matrix}\right] has. no. solutions, and Ax = \left[\begin{matrix}0 \\ 1\\0 \end{matrix}\right] \] has exactly one solution
(a) Give all posibble information about m and n and rank r of A
(b) Find all solutions to Ax = 0
(c) Write an example of a matrix A that fits the description in part (a).

Answer 5

snadig, if you review the lecture Strang on dimension, rank, and basis, he tells us that when r = n, that is when there is a pivot in every column, then the null space only contains the zero vector. This is because there are no free variables. So, in that case the solution only contains zero or one solution. In this case, he tells us the matrix (0,1,0) is a solution, so it must be in the column space. I think he does a better job explaining it in the lecture, but maybe that helps some.

Answer 6

datanewb, I agree with your explanation!

Answer 7

Thanks, I actually got confused about who was the OP. :) Now I see you only asked where to find the question, but thank you for letting me know my answer was sufficient.