Real vector space/Bijective/Orientation-preserving/Positive determinant/Fact/Proof

Proof

Let $v_{1}, \dots, v_{n}$ be a basis of $V$ . Because of the bijectivity of $φ$ , the images

φ (v_{1}), \dots, φ (v_{n})

form also a basis of

V

. Let

φ (v_{j}) = \sum_{i = 1}^{n} a_{i j} v_{i},

so that

M = {(a_{i j})}_{i j}

is the describing matrix of the mapping with respect to the basis $v_{1}, \dots, v_{n}$ . This matrix is also the base change matrix $M_{𝔳}^{φ (𝔳)}$ . The positivity of the determinant of this transformation matrix means by definition that the two matrices represent the same orientation.

To fact