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Imagine you're working with a tool that reshapes every figure you input, but maintains their basic relationships—scaling them up or down, flipping or turning them, yet always doing so consistently. In linear algebra, such a tool is called a linear map, and it has some special properties. Firstly, it's additive: when you map two figures together, it's the same as if you combined them first and then mapped them. Mathematically, this says that for any two inputs, let's call them 'shapes' x and y, a linear map f would satisfy the equation f(x + y) = f(x) + f(y).

Secondly, there's homogeneity: if you were to take one shape and enlarge or shrink it before mapping, the result is the same as mapping first and then resizing. In formula terms, this means for any shape x and any scalar (a resize factor) α, f(αx) = αf(x). These properties might sound abstract, but they will be pivotal in understanding why a figure, or a set in our case, keeps its fundamental characteristics, like convexity, even after being transformed.

A shape can be thought of as 'convex' if it's 'filled out'; imagine stretching a rubber band around it and letting go—it snaps to the edges without leaving any dents or gaps. In more formal terms, a set is convex if, when you choose any two points within it, every point on the straight line segment that joins them is also within the set. We can capture this idea using a mixture or weighted average of the two points: given any two points x and y in our set, for each 'mixing' value λ between 0 and 1, (1-λ)x + λy also lies in the set.

By making λ closer to 0, you're saying 'mostly x with a bit of y', and when λ is towards 1, it's the other way around; 'mostly y with a sprinkle of x'. If this works for every blend from all 'mostly x' to all 'mostly y', you've got yourself a convex set.

Now let's put our previously discussed concepts to the test. To show that the image of a convex set C stays convex under a linear map f, we pick any two images under the map, say, x' and y'. These are the 'after pictures' of two points from our convex set that we put through f. When we consider the line segment joining x' and y', we're blending them with that same λ 'mix' and want to see if the mixture is also an image.

Using our linear map properties, we see that the mixed result, z', can be expressed as f((1 - λ)x + λy). Since x and y come from our original convex set, their blend (1 - λ)x + λy, let's call it z, is in C, thanks to C's convexity. Since f is a map, and z is in C, f(z) must be in the 'after picture' set, C'. Thus, the mixture land, or point z', also lies in the image of our set under f, keeping the convexity intact.

Essentially, what we've shown is that a linear map, like a skilled sculptor, can reshape our original convex set into a new form, but it can't make it lose its fundamental convex 'fullness'. Very much like a shadow, which changes form with the light's angle, yet always resembles the original object.