 In this video, we are going to provide the solution to question number four from the practice midterm exam number two for math 2270 We are given a linear transformation t that goes from r2 to r3 and is given by the following rule t of x1 x2 The first coordinate will be x1 plus 4x2 You get 2x1 plus 5x2 for the second coordinate and 3x1 plus 6x2 for the third coordinate right there So what we have to do is we have to compute the change that the standard matrix Representation for this linear transformation. So this matrix right here is going to look like t of e1 and T of e2 that's what we have to compute for this question right here We're understanding that e1 is the vector 1 0 and e2 is the vector 0 1 so basically to find out the first column vector We're going to plug in 1 for x1 and then 0 for x2 and then you simplify that That's the first column and then for the second column Whoops for the second column you plug in as a 1 for x2 and you plug in a 0 For x1 you simplify that gives you the second column Personally though for me. I like it a lot better that I'm actually going to write this vector not as I Want to write it as a column vector not as a as a inline vector That is I don't want to write a horizontal I want to write it vertically And I want to match up all of the x's into a column all the x1s x2s into separate columns if you do that You get x1 plus 4x2 You're going to get 2x1 Plus 5x2 and then finally you get 3x1 plus 6x2 and when you write your when you write the transformation this way kind of like writing an augmented matrix to represent a System of linear equations you can basically see the coefficients in play right here So we see the coefficients from the first column are going to be 1 2 3 and for the second column You're going to see 4 5 6 I see what I did there now, and so that's what we're going to record here on the line The first column is 1 2 3 and then the second column is 4 5 6 and that is then the standard matrix representation of this linear transformation