Change of Basis. Row and Column Pictures for a Linear System
- Eugene Yablonski
An invertible system of two linear equations admits several geometric interpretations. 1. In the row picture (when the slider is in the left-most position), each equation is visualized as a line. The goal is to find the coordinates of the intersection point P. 2. In the column picture (when the slider is in the left-most position), each column of coefficients is a vector on a plane. The goal is to find a linear relationship among those vectors. It's easier to view the picture if you move the slider all the way to the right. Moving the slider changes just the viewer's frame but not the relative placement of vectors and lines. There are no such things as "absolute coordinates" in Linear Algebra! From this perspective, the row and column pictures are simply different visual interpretation of the same geometric phenomenon: change of coordinates.
QUESTIONS. 1. What are the - and -coordinates of the green vector ? 2. What are the - and -coordinates of the vector ? 3. What are the equations of the red coordinate axes in terms of the - and -coordinates? (See the answer in the attached pdf file).