Question

Prove that the vectors (2,-1,4), (1,-1,2), (3,1,-2) from a basis for R3​

Answer 1

Step-by-step explanation:

To show any given 3 vectors form a basis of R³ or not,

we need to show that these vecors are Linearly Independent and thesse vectors span the entire space R³.

Test for L.I:

Let us assume there exists non zero constants x and y such that

Let (2,1,4) = x(1,-1,2) + y(3,1,-2)

Now, comparing the components

x +3y = 2---(1),

-x +y = 1-----(2),

2x -2y =4---(3)

Looking at (2) and (3), x-y = -1 and x -y =2 which is impossible , hence there doesn't exist x,y satisfying the above relation.

Hence the vectors (2,1,4),(1,-1,2),(3,1,-2)  are linearly independent.

Since these are 3 in number,

note that Always any 3 number of Linearly independent vectors will span the entire space R³.

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