- #1
Joelly
- 3
- 0
In summary, the function that maps $a+b\sqrt{2}$ to $\begin{bmatrix}a & 2b \\ b & a \end{bmatrix}$ is an obvious "guess" and can be proven to be an isomorphism by showing that it is "one to one" and "onto" and that it "preserves the operation" for multiplication and addition. This is done by comparing the products and sums of two members in the first set and their corresponding mapped matrices.
- #2
HOI
- 921
- 2
An obvious "guess", practically forced by the way the problem is stated, is the function that maps $a+ b\sqrt{2}$ to $\begin{bmatrix} a & 2b \\ b & a \end{bmatrix}$. Is that an isomorphism? That is, is it "one to one", is it "onto", and does it "preserve the operation".
The first two are simple. To prove that this "preserves the operation" you need to take two members of the first set, $a+ b\sqrt{2}$ and $x+ y\sqrt{2}$, which map to $\begin{bmatrix}a & 2b \\ b & a\end{bmatrix}$ and $\begin{bmatrix}x & 2y \\ y & a\end{bmatrix}$ and show that the product $(a+ b\sqrt{2})(x+ y\sqrt{2})$ maps to the product $\begin{bmatrix}a & 2b \\ b & a \end{bmatrix}\begin{bmatrix}x & 2y \\ y & x\end{bmatrix}$.
Do those two multiplications and compare them.
To determine whether they are also isomorphic under addition, add those pairs and compare them.
The first two are simple. To prove that this "preserves the operation" you need to take two members of the first set, $a+ b\sqrt{2}$ and $x+ y\sqrt{2}$, which map to $\begin{bmatrix}a & 2b \\ b & a\end{bmatrix}$ and $\begin{bmatrix}x & 2y \\ y & a\end{bmatrix}$ and show that the product $(a+ b\sqrt{2})(x+ y\sqrt{2})$ maps to the product $\begin{bmatrix}a & 2b \\ b & a \end{bmatrix}\begin{bmatrix}x & 2y \\ y & x\end{bmatrix}$.
Do those two multiplications and compare them.
To determine whether they are also isomorphic under addition, add those pairs and compare them.
- #3
Joelly
- 3
- 0
What are some tips for eating healthy on a budget?
1. Plan your meals and make a grocery list: This will help you avoid buying unnecessary items and stick to healthy options.
2. Buy in bulk: Purchasing items in bulk can often save you money in the long run. Look for deals and discounts on healthy staple foods like grains, beans, and frozen fruits and vegetables.
3. Cook at home: Eating out can be expensive, so try to cook at home as much as possible. This allows you to control the ingredients and portion sizes, making it easier to eat healthy on a budget.
4. Buy seasonal produce: Fruits and vegetables that are in season are often cheaper and fresher than out-of-season options. Shop at local farmers' markets for the best deals.
5. Don't be afraid of frozen or canned options: Frozen fruits and vegetables are just as nutritious as fresh ones and can be more budget-friendly. Canned beans and vegetables are also affordable and can be a healthy addition to meals.
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