Linear combinations

Task number: 2509

In the vector space \(\mathbb R^4\) over the field \(\mathbb R\) find the linear combination of vectors \((-5{,}5,1,-1)^T\), \((2,-5{,}0,2)^T\), \((3{,}2,0,-2)^T\) a \((2,-3{,}1,1)^T\) which does lead to vector \((-7{,}12,2,-4)^T\). Is this linear combination unique?

  • Resolution

    It is possible to find coefficients of of the linear combination with use of a system of equations. We may obtain for example \((2{,}0,1{,}0)^T\) as a solution.

    The system of equations does not have a unique solution, thus the linear combination is not unique. The general form of the solution is \((2{,}0,1{,}0)^T+p(-1,-2,-1{,}1)^T\).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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