Math, asked by sourav1805, 11 days ago

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

Answers

Answered by hukam0685
4

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.

To learn more on brainly:

1) Two matrixes A and B are given. Express B^-1 through x and A.

https://brainly.in/question/41760349

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