Find the co-ordinate vectors of the vector with components (2,3,4,-1) relative to ordered basis {e1,e2,e3,e4}.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The requests of the problem is incomplete since it does not provides the vectors `bar e_1, bar e_2, bar e_3, bar e_4` .

Alleatory considering  `bar e_1 = (1,0,0,1), bar e_2 = (1,0,1,0)` , `bar e_3 = (1,1,0,0)` and `bar e_4= (0,0,1,1)` yields:

`(2,3,4,-1) = bar v_1*bar e_1 + bar v_2*bar e_2 + bar v_3*bar e_3 + bar v_4*bar e_4`

`(2,3,4,-1) = bar v_1*(1,0,0,1) + bar v_2*(1,0,1,0) + bar v_3*(1,1,0,0) + bar v_4*(0,0,1,1)`

`(2,3,4,-1) = (bar v_1 + bar v_2 + bar v_3, bar v_3, bar v_2 + bar v_4, bar v_1 + bar v_4)`

`{(bar v_1 + bar v_2 + bar v_3 = 2),(bar v_3 = 3),(bar v_2 + bar v_4 = 4),(bar v_1 + bar v_4 = -1):}`

`{(bar v_1 + bar v_2 = 2 - 3),(bar v_2 + bar v_4 = 4),(bar v_1 + bar v_4 = -1):} => {(bar v_1 + bar v_2 = -1),(bar v_2 + bar v_4 = 4),(bar v_1 + bar v_4 = -1):} => bar v_1 + bar v_2 = bar v_1 + bar v_4 => bar v_2 = bar v_4`

Since `bar v_2 + bar v_4 = 4` yields:

`bar v_2 + bar v_4 = 4 => 2bar v_2 = 4 => bar v_2 = bar v_4 = 2`

`bar v_1 = -1 - bar v_2 => bar v_1 = -1 - 2 => bar v_1 = -3`

Hence, evaluating the coordiate vector of `(2,3,4,-1)` yields `(bar v_1,bar v_2, barv_3, bar v_4) = (-3,2,3,2)` .

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