Consider the vectors vector u = < 1,-1, 3 >, vetor v = < 1, k, k >, vector w =< k, 1, k^2 >

For which values of k is {< 1,-1, 3 >,< 1, k, k >,< k, 1, k^2 >} a basis for R^3?"

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You need to consider the condition for the vectors `< 1,-1, 3 >,< 1, k, k >,< k, 1, k^2 >` to be linearly independent to form a basis on `R^3` , hence, you need to evaluate the following determinant, such that:

`Delta = [(1,-1,3),(1,k,k),(k,1,k^2)]`

The problem provides the information that the vectors form a basis on `R^3` , hence, `Delta != 0` , such that:

`Delta = k^3 + 3 - k^2 - 3k^2 - k + k^2`

`{(Delta = k^3 - 3k^2 - k + 3),(Delta != 0):} => k^3 - 3k^2 - k + 3 != 0`

`k^2(k - 3) - (k - 3) != 0 => (k - 3)(k^2 - 1) != 0`

`k - 3 != 0 => k != 3`

`k^2 - 1 != 0 => k^2 !=` ` => k != +-1`

**Hence, evaluating k for the vectors to form a basis on `R^3 ` yields that k may have any real value but **`k != {-1,3,1}.`

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