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I am referring to problem number 18 in problem set 1.1 of Introduction to Linear Algebra by Gilbert Strang.

Intuitively, I think that all the combinations $cv+dw$ for $0 \leq c \leq 1$ and $0 \leq d \leq 1$ would be the region inside the parallelogram which has $v$ and $w$ as its adjacent sides.

Because, if we keep d constant and vary c then the arrowheads of all the resultant vectors would lie along lines parallel to $v$.

Similarly, if we keep c constant and vary d then the arrowheads of all the resultant vectors would lie along lines parallel to $w$.

So, if we vary both, then we will get the entire region contained within the parallelogram.

But, I am unable to express it in concrete mathematical terms i.e. write it as a solid proof.

And that is my query, can I express it in concrete mathematical terms rather than intuitive understanding ?

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  • $\begingroup$ Geometrically, when do we say that a point lies inside the given parallelogram? $\endgroup$
    – Yathi
    May 12, 2022 at 8:34
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    $\begingroup$ I suspect that Strang is not hoping for something like this. I think he just wants you to think visually about what linear combinations mean, which is what you're doing. If you want formal proofs, first you need formal definitions, and to write the problem in those terms. You would have to decide what a parallelogram is, in the context of linear algebra, and without relying on ill-defined, intuitive words like "shape", "side", "in". Everything would have to be in terms of addition and scalar multiplication. I have a definition in mind, but it's pretty much what you're asking to prove here. $\endgroup$ May 12, 2022 at 9:13

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