Question

hi starting to learn linear algebra... solving Problem set 1.1. At problem one i see that [1,2,3] and [3,6,9] fill a line in R3 .. but why does just the line but not a plane ?? Thanks

Answer 1

Each one of those is the equation of a plane, by itself. In general, two equations of three variables can have either no intersection (parallel), intersection everywhere (ie., they are the same plane), or a line (the vast majority of cases).

Having gone through quite a bit of 18.06 myself, I wonder if you have taken 18.02, the multivariable calculus course? In particular, the early portion of the class focuses on vector mathematics and is definitely a prerequisite to this class. You don't need all of 18.02 though, mainly the vector mathematics. Although it's possible to brush up on it, maybe at the Khan academy perhaps.

Answer 2

These two vectors lies on the same line but that line lies on infinity many planes and these two vectors lies on infinity many planes

Answer 3

If you try to view those vectors geometrically you'll see that themselves and any of their scalar multiples lie on the same line. In fact one of them is a scalar multiple of the other one. And even more, all that you'll ever need to reprezent any combination of multiples of those vectors is just a line.
let A=[1,2,3]
let B=[3,6,9]
1)it's clear that B=3 * A, hence A and B share the same line (same direction only magnitude different, like 3 times longer)
2)it's clear that c*A, for any scalar c (rational or integer) is on the same line as A and B, just c times longer but in the same direction (i.e. line)
3)if you do C=A+B, you'll find that is again a multiple of A and B, also on the same line
All operations you can do with vectors that produce vectors are :
a)multiply by a scalar
b)add/subtract the 2 vectors and their multiples
given our 2 vectors here a line is sufficient to reprezent all you will ever get out of them by those 2 operations. Those 2 vector will never form more than a line.

Answer 4

Thank you every much everyone