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Relation between Variables and Equations

If rx y=0, th...

Question

If $r_{x y} = 0$ , the variable X and Y are

(a) linearly related

(b) not linearly related

(c) independent

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Solution

(b) If $r_{x y} = 0$ , it means the two variables are uncorrelated and there is no linear relation between them. However, other types of relationships may be there and they may not be independent.

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Similar questions

If r_xy= 0 the variable X and Y are

(i) linearly related

(ii) not linearly related

(iii) independent

A set of vectors

{(a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})}

is said to be linearly independent if and only if
$\begin{vmatrix}
a_{1} & a_{2} & a_{3}\\
b_{1} & b_{2} & b_{3}\\
c_{1} & c_{2} & c_{3}
\end{vmatrix}\neq 0$
otherwise the set is said to be linearly dependent. A similar result holds for

{(a_{1}, a_{2}), (b_{1}, b_{2})}

.

(a_{1}, a_{2}, a_{3})

(b_{1}, b_{2}, b_{3})

(c_{1}, c_{2}, c_{3})

are linearly independent and

x, y, z ϵ R

Q. Consider the following statements about the linear dependence of the real valued functions

y_{1} = 1, y_{2} = x

y_{3} = x^{2}

, over the field of real numbers.

I.

y_{1}, y_{2}

y_{3}

are linearly independent on

- 1 \leq x \leq 0

y_{1}, y_{2}

y_{3}

are linearly dependent on

0 \leq x \leq 1

y_{1}, y_{2}

y_{3}

are linearly dependent on

0 \leq x \leq 1

y_{1}, y_{2}

y_{3}

are linearly independent on

- 1 \leq x \leq 1

Which on among the following is correct?

a, b, c

d

are linearly independent set of vectors and

K_{1} a + K_{2} b + k_{3} C + K_{4} d = 0

Q. Assertion :If

\to a =^i, \to b =^j

\to c =^i +^j

\to a

\to b

are linearly independent but

\to a, \to b

\to c

are linearly dependent. Reason: If

\to a

\to b

are linearly dependent and

\to c

is any vector then

\to a, \to b

\to c

are linearly dependent.

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