I am going through Gilbert Strang's book Introduction to Linear Algebra Edition 4. I need help with one of the questions in the exercise.
It asks to draw the line of all the combinations $cv + dw$ such that $c + d = 1$. I am able to verify that all such linear combinations of v and w indeed lie on the line passing through v and w in the following way:
Let the tail point of v be $(x_1, y_1)$ and w be $(x_2, y_2)$.
So, the equation of the line would be
$ y – y_1 = \frac{y_2 – y_1}{x_2 – x_1}(x – x_1)$
It can be verified that the point $(cx_1 + dx_2, cy_1 + dy_2)$ satisfies the equation of the line. But is there any geometric meaning or intuition through which it can be realized that such linear combinations would lie on this line?
This shows the linear combination $u$ when $c = d = 1/2$
Best Answer
One is right there in the diagram in your question. From the parallelogram rule we can see that the dotted segment corresponds to $\mathbf d = \mathbf w-\mathbf v$ (or $\mathbf v-\mathbf w$, depending on the direction you’re going). The line consists of the points that can be reached by adding some multiple of $\mathbf d$ to $\mathbf v$, i.e., $\mathbf v+\lambda(\mathbf w-\mathbf v) = (1-\lambda)\mathbf v+\lambda\mathbf w$. What this says is that whatever proportion of $\mathbf w$ you’re adding, to stay on the line you have to take away that same proportion of $\mathbf v$, and so the coefficients stay balanced.
Another way to look at it is that the line through $\mathbf v$ and $\mathbf w$ is the image of the line $x+y=1$ under a linear transformation that maps $(1,0)$ to $\mathbf v$ and $(0,1)$ to $\mathbf w$. This transformation maps the point $(x,y)$ to $x\mathbf v+y\mathbf w$, but you’ve still got the constraint that $x+y=1$: because of the linearity of the transformation, that doesn’t change.
Incidentally, this is called an affine combination of $\mathbf v$ and $\mathbf w$. You’ll encounter these with more points or vectors in other contexts as well.