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If

`B`

is the standard basis of the space

`P_(3)`

of polynomials, then let

`B={1,tt^(2),t^(3)}`

. Use coordinate vectors to test the linear independence of the set of polynomials below. Explain your work.

`1-7t^(2)-t^(3),t+6t^(3),1+t-7t^(2)`

Write the coordinate vector for the polynomial

`1-7t^(2)-t^(3)`

.

`(1,0,-7,-1)`

Write the coordinate vector for the polynomial

`t+6t^(3)`

.

`(0,1,0,6)`

Write the coordinate vector for the polynomial

`1+t-7t^(2)`

.

`(1,1,-7,0)`

To test the linear independence of the set of polynomials, row reduce the matrix which is formed by making each coordinate vector a column of the matrix. If possible, write the matrix in reduced echelon form.

`[[1,0,1],[0,1,1],[-7,0,-7],[-1,6,0]]∼`