Question

PROVE THAT THE VECTORS (2,-1,4),(1,-1,2),(3,1,2) FROM A BASIC FOR R3.

Answer 1

Answer:

Step-by-step explanation:

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

we need to show that these vectors are Linearly Independent and these 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.

Read more on Brainly.in - https://brainly.in/question/7237172#readmore