Linear Combinations in 2-space

[Destroxia]

Homework Statement

In the xy-plane mark all nine of these linear combinations:

$α \lbrack 2, 1 \rbrack + β \lbrack 0, 1 \rbrack$ with c = 0, 1, 2 and d = 0, 1, 2

Homework Equations

ANSWER:

The nine combinations will lie on a lattice. If we took all whole numbers c and d, the lattice would
lie over the whole plane.

The Attempt at a Solution

I think my biggest problem in this, is not actually knowing what the question is asking.

I've tried plotting the linear combinations of me filling in α and β with the numbers listed for c & d.

1. $0 \lbrack 2, 1 \rbrack + 0\lbrack 0, 1 \rbrack$ yields, $\lbrack 0, 0 \rbrack$

2. $1 \lbrack 2, 1 \rbrack + 1\lbrack 0, 1 \rbrack$ yields, $\lbrack 2, 2 \rbrack$

3. $2 \lbrack 2, 1 \rbrack + 2\lbrack 0, 1 \rbrack$ yields, $\lbrack 4, 4 \rbrack$

I'm not sure where they are getting 9 linear combinations, but I'm pretty sure I'm misunderstanding the instructions of the problem.

 

[vela]

Hint: $3 \times 3 = 9$

 

[Destroxia]

Hint: $3 \times 3 = 9$

Okay, so the only thing I could think of using that hint would be that I can do other permutations of the c = 0, 1, 2 and d = 0, 1, 2?

So maybe do 1 and 0, or 2 and 1, etc?

 

[Mark44]

Homework Statement

In the xy-plane mark all nine of these linear combinations:

$α \lbrack 2, 1 \rbrack + β \lbrack 0, 1 \rbrack$ with c = 0, 1, 2 and d = 0, 1, 2

Seems like it shoud be "with α = 0, 1, and 2, and β = 0, 1, and 2"

BTW, it's much simpler to just use the [ and ] characters than typing lbrack and rbrack in LaTeX.

Homework Equations

ANSWER:

The nine combinations will lie on a lattice. If we took all whole numbers c and d, the lattice would
lie over the whole plane.

The Attempt at a Solution

I think my biggest problem in this, is not actually knowing what the question is asking.

I've tried plotting the linear combinations of me filling in α and β with the numbers listed for c & d.

1. $0 \lbrack 2, 1 \rbrack + 0\lbrack 0, 1 \rbrack$ yields, $\lbrack 0, 0 \rbrack$

2. $1 \lbrack 2, 1 \rbrack + 1\lbrack 0, 1 \rbrack$ yields, $\lbrack 2, 2 \rbrack$

3. $2 \lbrack 2, 1 \rbrack + 2\lbrack 0, 1 \rbrack$ yields, $\lbrack 4, 4 \rbrack$

I'm not sure where they are getting 9 linear combinations, but I'm pretty sure I'm misunderstanding the instructions of the problem.

 

[Mark44]

Okay, so the only thing I could think of using that hint would be that I can do other permutations of the c = 0, 1, 2 and d = 0, 1, 2?

So maybe do 1 and 0, or 2 and 1, etc?

Each choice of c (really $\alpha$) can be paired with one of three possible d (really $\beta$) values. How many combinations of the two does that make?

 

[Destroxia]

Each choice of c (really $\alpha$) can be paired with one of three possible d (really $\beta$) values. How many combinations of the two does that make?

Yeah, that's what I was getting at. I didn't know if the points could be interchanged, but I guess there isn't really any reason why they couldn't be.