Question

prove that the vectors (2 1 4) (1 -1 2) 3 1 -2) for a basic of be(R).​

Answer 1

Step-by-step explanation:

Given:Prove that vectors (2,1,4), (1,-1,2),(3,1,-2) form a basis of V³(R).

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≠0

Therefore,

Vectors are linearly independent thus form a basis of V³(R)

Thus,

It has been prove that vectors (2,1,4), (1,-1,2),(3,1,-2) form a basis of V³(R).

Hope it helps you.

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