Is there a reason that linear maps have to keep the origin fixed?
keenan
Is there a reason that linear maps have to keep the origin fixed?
That's the definition: a map is said to be "linear" if it preserves lines through the origin.
One way to understand why that's a useful definition is the discussion on the next slide: if a map preserves lines through the origin (i.e., if it's linear), then it preserves vector space operations. I.e., addition and scalar multiplication before and after the map are equivalent. That's a property you really don't want to break, because all of a sudden you can no longer do manipulations like $A(x+b) = Ax + Ab$ (when using a matrix representation, say). Would be a very sad day indeed...
haoala
Was the result of the non-linear map shown on this slide generated by a particular function on the pentagon, or is it just an arbitrary Powerpoint shape to convey the idea?
keenan
Was the result of the non-linear map shown on this slide generated by a particular function on the pentagon
IIRC it was some combination of bending and twisting in Illustrator. So, somewhere deep inside Adobe's code, there is probably a closed-form function that describes this transformation. But no, I have no clue what it is.
Here's a more interesting question: suppose I show you the nonlinear deformation of the pentagon above. I.e., I just hand you the initial boundary curve and the final boundary curve. Can you come up with a function that maps the first to the second? One way you could start to approach this problem:
First, find a mapping from the boundary to the boundary (i.e., just figure out how the black line gets deformed).
Next, "extend" this mapping to the interior, i.e., find a mapping from the whole pentagon to the whole deformed pentagon that agrees with your map along the boundary, and is somehow nice and smooth on the interior.
There are many, many different ways people have approached this problem, with a lot of very fun and mathematically interesting solutions. Here's a small sampling (could make a good final project!):
When we say that linear maps have a fixed origin, does it necessarily have to be symmetric about the origin? Also, isn't "origin" an arbitrary value that we decide for reference?
Building off of that, if its important that the origin has to be fixed, can we translate it to the origin and then perform linear maps? Or does that quality to affine maps alone?
Is there a reason that linear maps have to keep the origin fixed?
That's the definition: a map is said to be "linear" if it preserves lines through the origin.
One way to understand why that's a useful definition is the discussion on the next slide: if a map preserves lines through the origin (i.e., if it's linear), then it preserves vector space operations. I.e., addition and scalar multiplication before and after the map are equivalent. That's a property you really don't want to break, because all of a sudden you can no longer do manipulations like $A(x+b) = Ax + Ab$ (when using a matrix representation, say). Would be a very sad day indeed...
Was the result of the non-linear map shown on this slide generated by a particular function on the pentagon, or is it just an arbitrary Powerpoint shape to convey the idea?
IIRC it was some combination of bending and twisting in Illustrator. So, somewhere deep inside Adobe's code, there is probably a closed-form function that describes this transformation. But no, I have no clue what it is.
Here's a more interesting question: suppose I show you the nonlinear deformation of the pentagon above. I.e., I just hand you the initial boundary curve and the final boundary curve. Can you come up with a function that maps the first to the second? One way you could start to approach this problem:
There are many, many different ways people have approached this problem, with a lot of very fun and mathematically interesting solutions. Here's a small sampling (could make a good final project!):
When we say that linear maps have a fixed origin, does it necessarily have to be symmetric about the origin? Also, isn't "origin" an arbitrary value that we decide for reference? Building off of that, if its important that the origin has to be fixed, can we translate it to the origin and then perform linear maps? Or does that quality to affine maps alone?