# Is null(T-\lambdaI) invariant under S for every \lambda \in F?

• pte419
In summary, proving null is invariant means demonstrating that the null hypothesis remains unchanged by a specific change or intervention. This is typically determined through statistical tests, such as ANOVA or t-tests. It is important to prove null is invariant to ensure the validity and reliability of research findings. However, there are limitations, such as differences in contexts or populations, as well as the influence of statistical tests and sample size. Some practical applications of proving null is invariant include evaluating the effectiveness of treatments or interventions, assessing policy changes, and identifying biases in research studies.
pte419
Suppose S, T $$\in$$ L(V) are such that ST=TS. Prove that null(T-$$\lambda$$I) is invariant under S for every $$\lambda$$ $$\in$$ F.

I can't find anything helpful in my book, I'm not sure where to start...

null(T-lambda*I) is just the set of all vectors x such that (T-lambda*I)x=0. So Tx-lambda*x=0. Or Tx=lambda*x. To show this is invariant under S you just have to show that if x satisfies that property, then so does Sx.

## 1. What does it mean to "prove null is invariant"?

To prove null is invariant means to demonstrate that the null hypothesis remains unchanged or unaffected by a particular change or intervention.

## 2. How is the invariance of null hypothesis determined?

The invariance of null hypothesis is typically determined by conducting statistical tests, such as ANOVA or t-tests, which compare the null hypothesis to alternative hypotheses and determine if there is enough evidence to reject or fail to reject the null hypothesis.

## 3. Why is it important to prove null is invariant?

Proving null is invariant is important because it helps to ensure the validity and reliability of research findings. If the null hypothesis is not invariant, it can lead to incorrect conclusions and potentially misleading results.

## 4. Are there any limitations to proving null is invariant?

Yes, there are limitations to proving null is invariant. It is possible that the null hypothesis may be invariant in one context or population, but not in another. Additionally, the results may also depend on the specific statistical test used or the sample size.

## 5. What are some practical applications of proving null is invariant?

Proving null is invariant can have practical applications in various fields, such as medicine, social sciences, and engineering. For example, it can help to determine the effectiveness of a new treatment or intervention, or to assess the impact of a policy change. It can also aid in identifying potential biases in research studies.

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