# Is Answer to Subspace Question Acceptable?

• jackdamack10
In summary: Third condition: Consider the vector \vec{v}=(v_1,v_2,v_3) in S. According to the definition of scalar multiplication in \mathbb{R}^3, we have \vec{v}=\vec{u}_1*\vec{u}_2. But \vec{u}_1 and \vec{u}_2 are in S, so \vec{v}=\vec{u}_1*\vec{u}_2+\vec{u}_3. According to the definition of addition in \mathbb{R}^3, we have \vec{v}+\vec{
jackdamack10
I just wanted to know if my answer is acceptable.

Q: S={(x,y,z) E $\mathbb{R}^{3}$ l x^2 + y^2 +z ^2 =0}
Is it a subspace of $\mathbb{R}^{3}$?

It is a subspace if x=0, y =0, z= 0

Let u=(0,0,0) u2=(0,0,0) and k be a scalar

u + u2 = (0,0,0) Closed under addition

ku = k(0,0,0) = (0,0,0) Closed under scalar multiplication

Is my answer complete? I'm not sure if I'm allowed to assume that the values of x,y,z are 0.

Thank you in advance

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jackdamack10 said:
Is my answer complete? I'm not sure if I'm allowed to assume that the values of x,y,z are 0.

Actually, S={(x,y,z) E $\mathbb{R}^{3}$ l x^2 + y^2 +z ^2 =0} = {(0,0,0)}

I.e. (0,0,0) is the only element in that set. It is made clear if you recall that x²+y²+z²=R² describe a sphere of radius R. Here R=0. The only point "on" a sphere of radius 0 is (0,0,0).

So, with this in mind, instead of the tentative "It is a subspace if x=y=z=0", start the proof with: "The only element of S is (0,0,0)", and then go on to show that the 3 conditions for S to be a subspace of R^3 are satisfied (like you did).

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Ok thanks.

What if I have a multiplication:

S={(x,y,z) E R^3 l xy =0}.
To prove that this is a subspace (or not a subspace), the only way I know to do so is by assuming a value for x, y and z before checking if it's closed under scalar mutiplication adn addition. Is there another way?

You can use $$a=(a_1,a_2,a_3), b=(b_1,b_2,b_3)$$ and since either x or y must be 0 to satisfy your condition of xy=0, you can have the condition that either $$a_1$$ or $$a_2$$ must be zero, and similarly for $$b_1$$ and $$b_2$$. Then show that a and b are closed under scalar addition and multiplication, and you have proofed it for all of the values in the subspace

jackdamack10 said:
Ok thanks.

What if I have a multiplication:

S={(x,y,z) E R^3 l xy =0}.
To prove that this is a subspace (or not a subspace), the only way I know to do so is by assuming a value for x, y and z before checking if it's closed under scalar mutiplication adn addition. Is there another way?

You must never assume a particular value of (x,y,z). You must always work your ways through the 3 conditions while assuming the most general form possible for (x,y,z).

First condition: Is (0,0,0) in S? Consider (x=0,y=0,z=0). Then xy=0, so (0,0,0) is in S. *check*

Second condition: Consider two vectors $\vec{u}_1 = (x_1,y_1,z_1)$ and $\vec{u}_2 = (x_2,y_2,z_2)$ in S. Because they are in S, they have the property that $x_1 y_1 = 0$ and $x_2 y_2 = 0$. According to the definition of addition in $\mathbb{R}^3$, we have $\vec{u}_1+\vec{u}_2 = (x_1+x_2,y_1+y_2,z_1+z_2)$. Now the condition: Does $(x_1+x_2)(y_1+y_2)=0$?. Let's see: In view of the "axiom of distributivity" in $\mathbb{R}$, $(x_1+x_2)(y_1+y_2)=x_1y_1+x_1y_2+x_2y_1+x_2y_2$. We know by hypothesis that $x_1y_1 = x_2y_2=0$. But what about the other two terms? It could be that $x_1\neq 0, \ y_1 = 0, \ x_2 = 0, \ y_2 \neq 0$. In this case, $x_1y_1 = 0$ and $x_2y_2 = 0$ are indeed satisfied but $x_1y_2 \neq 0$. Conclusion: $\vec{u}_1+\vec{u}_2 \notin S \ \forall \vec{u}, \ \vec{u}_2 \ \Rightarrow$ S is not a subspace of $\mathbb{R}^3$ because condition 2 (closed under addition) is not met.

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## 1. What is a subspace question?

A subspace question is a type of question that asks for a specific subset or portion of a larger set of data or information. It is commonly used in statistics and mathematics to help narrow down and analyze data.

## 2. How can I determine if an answer is acceptable for a subspace question?

The acceptability of an answer for a subspace question depends on the specific criteria and conditions set by the person asking the question. Generally, an acceptable answer should meet the requirements and limitations of the subspace question and provide relevant and accurate information.

## 3. Are there any limitations to using subspace questions?

Yes, there are limitations to using subspace questions. These questions can only be used for specific types of data and may not be applicable in all situations. Additionally, the accuracy and reliability of the answers can be affected by the quality and quantity of the data used in the analysis.

## 4. Can subspace questions be used in all fields of science?

While subspace questions are commonly used in mathematics and statistics, they can also be applied in other fields of science such as biology, physics, and social sciences. However, their applicability may vary depending on the type of data and research being conducted.

## 5. How can I improve the accuracy of my subspace question analysis?

To improve the accuracy of subspace question analysis, it is important to carefully select and define the criteria and conditions for the question. Additionally, using a larger and more diverse dataset can also help improve the accuracy of the results. It is also important to critically evaluate the results and consider any potential biases or limitations in the analysis.

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