System of vectors $a_{1}, a_{2}, . . . . . . . a_{n}$ is said to be linearly dependent if there exists a system of scalars $(c_{1}, c_{2} - - - c_{n} s u c h t h a t c_{1} ¯ a + c_{2}_{2} + . . . c_{n}_{n} = ¯ 0$

True

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False

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Solution

The correct option is B False
Please read the given statement carefully. Are all the conditions required for linear dependence mentioned? No. Vectors $a_{1}, a_{2} . . . . . a_{n}$ are linearly dependent if there exists a system of scalars $c_{1}, c_{2} . . . . c_{n}$ such that $c_{1}_{1} + c_{2}_{2} + . . . c_{n}_{n} = ¯ 0$ But there is a condition on scalars that all of them can’t be zero simultaneously. At least one of them has to be non-zero. This is missing in the given statement. So it’s false.

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Similar questions

Q. System of vectors

a_{1}, a_{2}, . . . . . . . a_{n}

is said to be linearly dependent if there exists a system of scalars

(c_{1}, c_{2} - - - c_{n} s u c h t h a t c_{1} ¯ a + c_{2}_{2} + . . . c_{n}_{n} = ¯ 0

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})

and

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

are linearly independent and

x, y, z ϵ R

, then

If $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ for the system of equations $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$

Q. A system of vectors is said to be coplanar, if
I. Their scalar triple product is zero.
II. They are linearly dependent.
Which of the following is true?

Q. Let n be an odd integer greater than 1 and let

c_{1}, c_{2}, \dots, c_{n}

be integers. For each permutation

a = (a_{1}, a_{2}, \dots, a_{n})

(1, 2, \dots, n)

define

S (a) = n \sum i = 1 c_{i} a_{i} .

Prove that there exist permutations

a \neq b

{1, 2, \dots, n}

such that n! is a divisor of S(a)-S(b).

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System of vectors a1,a2,.......an is said to be linearly dependent if there exists a system of scalars (c1,c2−−−cn such that c1¯a+c2¯a2+...cn¯an=¯0

System of vectors $a_{1}, a_{2}, . . . . . . . a_{n}$ is said to be linearly dependent if there exists a system of scalars $(c_{1}, c_{2} - - - c_{n} s u c h t h a t c_{1} ¯ a + c_{2}_{2} + . . . c_{n}_{n} = ¯ 0$