This is to check some of my work, and to get some additional help.
I will denote a matrix as I would in Maple A = ([1,2],[3,4],[4,5]), for example, will mean a 3x2 matrix, with the first row being [1,2].
1.) Given that
Matrix A = ([1,3],[2,1],[7,2]) and that
vector b = (, , )
a.) Is A*x = b consistent or inconsistent?
I row reduced it to echelon form, and got:
Therefore, since there is a pivot in the augmented column, the solution is inconsistent, that is, has no solutions.
b.) Draw a sketch of the vectors in R^3 that you'll illustrate your answer in part a. Note: this doesn't have to be a 'to scale' graph, instead, just a general picture of what's happening, with a paragraph or two. (HINT: think in terms of the span of columns of A).
Inconsistent? Graph exists?
c.) Now, draw a sketch of the lines in R^2 that illustrates the answer in part a. And since since you'll be drawing this in R^2, this should be easier to graph and should be done 'more to scale'. (HINT: Think in terms of orig. system of lin. equations.)
Again, if it's inconsistent, how are you able to draw a sketch of part a? I was thinking along the lines of there being a free variable, that is the R generated, in R^3, but this isn't the case.
2.) Let g be the transformation from R^3 to R^3 defined as:
g : R^3 -> R^2 with
vector ([x_1],[x_2],[x_3]) |-> matrix ([x_1 + x_2 + x_3],[x_1 - 2])
Is g a linear transformation? If it is, show why it is. If it is not, show why it is not.
I believe this involves some of the principles T(u + v) = T(u) + T(v) or something to that extent, though I'm not too sure how to go about this.