Why do we care if a set forms a basis?

Which of the following statements is not correct for the relation R defined by aRb, if and only if b lives within one kilometre from a?
A) R is reflexive
B) R is symmetric
C) R is not anti-symmetric
D) None of the above

A line segment is a part of a line as well as a ray. True or False

Which characteristic of a data set makes a linear regression model unreasonable?

Find the meaning of 'Sxx' and 'Sxy' in simple linear regression

In the least-squares regression line, the desired sum of the errors (residuals) should be
a) zero
b) positive
c) 1
d) negative
e) maximized

Can the original function be derived from its $k^{th}$ order Taylor polynomial?

Should the independent (or dependent) variables in a linear regression model be normal or just the residual?

What is the relationship between the correlation of two variables and their covariance?

What kind of technique is to be adopted if I have to find an equation or model for say, $D$ depends on $C$, $C$ changes for a set of $B$, which changes for different $A$.

Correlation bound
Let x and y be two random variables such that:
Corr(x,y) = b, where Corr(x,y) represents correlation between x and y, b is a scalar number in range of [-1, 1]. Let y' be an estimation of y. An example could be y'=y+(rand(0,1)-0.5)*.1, rand(0,1) gives random number between 0, 1. I am adding some noise to the data.
My questions are:
Is there a way where I can bound the correlation between x, y' i.e. Corr(x,y')?I mentioned y' in light of random perturbation, I would like to know what if I don't have that information, where I only know that y' is a estimation of y. Are there any literature that cover it?

What is the benefit of OU vs regression for modeling data, say data in the form of ($x,y$) pairs?

Can you determine the correlation coefficient from the coefficient of determination?

How can one find the root of sesquilinear form with positive definite matrix?

From numerical simulation and regression analysis I discovered that the root-mean-square amplitude of white noise with bandwidth $\mathrm{\Delta <!--\; \Delta \; -->}f$ is proportional to $\sqrt{\mathrm{\Delta <!--\; \Delta \; -->}f}$. How can this be derived mathematically ?

In a Simple Linear Regression analysis, independent variable is weekly income and dependent variable is weekly consumption expenditure. Here $95$% confidence interval of regression coefficient, ${\mathrm{\beta <!--\; \beta \; -->}}_1$ is $(.4268,.5914)$.