Suppose [tex]T: \mathbb{R}^n \to \mathbb{R}^n[/tex] is invertible.

(a) Is it necessarily one-to-one?

(b) Is it necessarily onto [tex]\mathbb{R}^n[/tex]?

Justify your answers.

(a) If T:Rⁿ→Rⁿ is invertible, then it is necessarily one-to-one. This means that for every vector in the domain, there is a unique vector in the range that it maps to. The invertibility of T guarantees that no two distinct vectors in the domain will be mapped to the same vector in the range.

(b) If T:Rⁿ→Rⁿ is invertible, then it is necessarily onto Rⁿ. This means that for every vector in the range, there is a vector in the domain that maps to it. The invertibility of T guarantees that every vector in the range is reachable by some vector in the domain.

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Suppose [tex]T: \mathbb{R}^n \to \mathbb{R}^n[/tex] is invertible.(a) Is it necessarily one-to-one?(b) Is it necessarily onto [tex]\mathbb{R}^n[/tex]?Justify your answers.

Answer :

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Explanation:

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Other Questions

Suppose [tex]T: \mathbb{R}^n \to \mathbb{R}^n[/tex] is invertible.

(a) Is it necessarily one-to-one?

(b) Is it necessarily onto [tex]\mathbb{R}^n[/tex]?

Justify your answers.