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Prove that the vectors a, b, c form a basis, and find the coordinates of the vector d in this basis.
Let R³ with respect to the canonical bases given four vectors f ₁ = (1,2,3), f₂ = (2,3,7),f₃ =(1,3,1), x = (2,3,4). Prove that the vectors f₁, f₂, f₃ can be taken as the new basis. Find coordinates η₁, η₂, η₃ of x with respect to this base.

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Decision (download format Word):
Write the transition matrix À:

and find its determinant
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We see that the rank of C is equal to three. From the theorem of the base minor vectors f ₁ , f, f₃ are linearly independent, and therefore can be taken as the basis for the space R³.
Find the inverse matrix À^-1.
A transparent matrix

Cofactors
                    
                   
                   
                   
                    
                   
                   
                  
                   
Inverse matrix À^-1

Find the coordinates of x with respect to the new basis.

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