Vault Of Euclid

Image and Preimage of the Unit Circle Under a Linear Map

Linear transformations core

Consider the mapping $F: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $F(x,y) = (3y, 2x)$. Let $S$ be the unit circle in $\mathbb{R}^2$, that is, the solution set of $x^2 + y^2 = 1$. (a) Describe $F(S)$. (b) Find $F^{-1}(S)$.
linear-transformationsimage-of-a-setpreimage-of-a-setcoordinate-geometryellipseunit-circlelinear-algebrafind-imagefind-preimage

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What this problem tests

Tests the understanding of the definitions of image and preimage of sets, and the ability to manipulate algebraic equations to identify geometric shapes.

This is a standard problem in introductory linear algebra, illustrating how linear transformations map geometric shapes like circles to ellipses.

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