Question

In the first lecture Prof Strang shows the two vectors (columns) of two linear equations in a visual representation. In this column picture he maps two x values onto an x,y space, where one x now becomes y. He does the same for the two y values, where one y becomes an x. Why is this a valid way of modeling. I understand everything after and how vectors work but this always seemed an odd leap.

Answer 1

This example may be confusing because the dimensions of the vectors is the same as the number of unknowns, namely a dimension equal to 2. The two components of each column vector do not correspond to the two unknowns in the equations. The equations naturally lend themselves to a row vector representation, but by viewing the system of equations as scalar multiples of column vectors on the left hand side (LHS) equal to a column vector on the right hand side (RHS), solving for the unknowns amounts to figuring out what scalar multiples of the column vectors on the LHS produces the column vector on the RHS. The key point is that by looking at a system of linear equations in this fashion, the column picture gives us deep insight about the nature of the solution(s) to such linear system, a cleaner geometrical interpretation/visualization (vectors in 3D are easier to draw and think about in comparison to intersecting planes - especially when drawn on a 2D medium), and a natural extension to n-dimensional space i.e. vectors with n components which we can longer draw or visualize. Now, more direct to your question, to clarify any ambiguity, think of the coordinate axes for the column vector as u and v instead of x and y, for example. The first component of the column vector is the u component, the second is the v component. If the vector is [1,2], then x just scales this to [1*x,2*x], and likewise, y will scale any other vector [a,b] in the u-v coordinates, to [y*a,y*b]. It may also help to add one more equation so now your system has three equations and two unknowns. In this case, your column vectors have three components (say along u,v, and w) and you still have only two unknowns, x and y which scale each column vector. Now it may become more obvious that the x and y are no longer thought of as coordinates in the 3D space when looking at the column picture, but are simply scalar coefficients, unknown prior to solving the system.

Answer 2

Thank you this is a wonderful answer and helps me a lot!