Question #218164
Let v1= (1,0,-1), v2=(1,1,1) , v3= (2,2,3). Find the dual basis for {v1,v2,v3).
1
Expert's answer
2021-07-19T17:17:30-0400

By filtering the given vectors into a matrix we have



B = [112012113]\begin{bmatrix} 1 & 1 &2\\ 0 & 1&2\\ -1&1&3 \end{bmatrix}


B-1 = [110252121]\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



γ1\gamma_1 = [1, -2, 1]


γ2\gamma_2 = [-1, 5, -2]


γ3\gamma_3 = [0, -2, 1]


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



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