Question

Prove that the vectors (2, 1, 4), (1, –1, 2) and (3, 1, –2) for a basis of V3
(R).

Answer 1

Step-by-step explanation:

Given:

(2, 1, 4), (1, –1, 2) and (3, 1, –2)

To find:Prove that these vectors for a basis of V³(R).

Solution:

To prove given vectors are the basis for V³(R),we have to prove that all three vectors are linearly independent.

If determinant of all three is not equal to zero,then one can say that all these vectors are independent.

Let

$\vec a=2i+j+4k\\\vec b=i-j+2k\\\vec c=3i+j-2k$

Put these values in determinant

$\left|\begin{array}{ccc}2&1&4\\1&-1&2\\3&1&-2\end{array}\right|$

Expand the determinant along R1

=2(2-2)-1(-2-6)+4(1+3)

=0-1(-8)+4(4)

=8+16

=24

Determinant ≠ 0

Therefore,one can conclude that,

Vectors are linear independent thus form a basis for V³(R).

Hope it helps you.

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