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Question
Any help would be greatly appreciated. I a having difficulty understanding this problem.

Take an m by n matrix A. Let vector b be a vector in R^m. Assume that

vector x = <a_1,...,a_n>

is a solution for the matrix equation A(vector x) = vector b. Show that vector b is in the column space of A by writing vector b as a linear combination of the columns of A.

If vector b is in R^m, then vector b = <b_1, ..., b_m>.
I will write that as b[m].

It is given that vector x = <a_1, ..., a_n>.
To keep it as x and not confuse it with A, I will make it x[n] = <x_1, ..., x_n>,

The matrix A is defined to be m x n, so write matrix A as A[m,n].

From this, we have A[m,n] * x[n] = b[m].

Take the ith row of matrix A as A[i].

From this, it can be seen that A[i,j] = A[i]*b[j].

Using this notation, it can be seen that the result of the multiplication is given as a linear combination of the rows of A times the elements of x.

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#### Scott A Wilson

##### Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

##### Experience

Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.