Answer to Question #218164 in Linear Algebra for Dhruv rawat

Question

Let v1= (1,0,-1), v2=(1,1,1) , v3= (2,2,3). Find the dual basis for {v1,v2,v3).

Answer 1

By filtering the given vectors into a matrix we have

B = $\begin{bmatrix} 1 & 1 &2\\ 0 & 1&2\\ -1&1&3 \end{bmatrix}$

B^-1 = $\begin{bmatrix} 1 & -1 &0\\ -2 & 5&-2\\ 1&-2&1\\ \end{bmatrix}$

Now the product B^-1 B = I

Hence we see that the the rows of B^-1 multiplied by the columns of B satisfy the condition for duality.

Hence, defining

$\gamma_1$ = [1, -2, 1]

$\gamma_2$ = [-1, 5, -2]

$\gamma_3$ = [0, -2, 1]

form the dual basis for {v₁, v₂, v3} .