Not every linear transformation has an inverse. Let’s break down why that is.
For a transformation to have an inverse, it needs to meet two important conditions:
Injective: This means that each input (or starting point) goes to a different output (or ending point). If different inputs end up with the same output, we can’t go back.
Surjective: Here, every possible output must come from some input. If there are outputs that don’t correspond to any input, we can’t reach them from the starting points.
If a linear transformation doesn't satisfy both of these conditions, it won’t have an inverse.
A common example is when we squish dimensions. Imagine taking a point in 3D space and moving it to a flat 2D plane. In this case, many 3D points can end up in the same spot on the 2D plane. We lose some information, so we can’t go back to the original 3D point.
So, remember to always check if a transformation is injective and surjective to see if it has an inverse!
Not every linear transformation has an inverse. Let’s break down why that is.
For a transformation to have an inverse, it needs to meet two important conditions:
Injective: This means that each input (or starting point) goes to a different output (or ending point). If different inputs end up with the same output, we can’t go back.
Surjective: Here, every possible output must come from some input. If there are outputs that don’t correspond to any input, we can’t reach them from the starting points.
If a linear transformation doesn't satisfy both of these conditions, it won’t have an inverse.
A common example is when we squish dimensions. Imagine taking a point in 3D space and moving it to a flat 2D plane. In this case, many 3D points can end up in the same spot on the 2D plane. We lose some information, so we can’t go back to the original 3D point.
So, remember to always check if a transformation is injective and surjective to see if it has an inverse!