the 'unit funcion' in '1v=v' is function equal to 1 for all x?
xinhez
If functions are vectors, how would we define the direction and magnitude? (To be pedantic).
vasua
I don't think it's necessary to define a direction and magnitude in the conventional sense for functions. If I'm not mistaken, a vector is simply an element of a vector space which only needs to satisfy the properties above, and since a function satisfies these constraints, it is a vector.
That said, think about what the direction of a standard vector represents. The values are simply coefficients of the corresponding bases vectors in a summation, e.g. (x, y) is just x * (1, 0) + y * (0, 1) with the standard bases. So, the direction is just a measure of components along each basis vector, which happens to have a direct spatial meaning when working with 2-space and 3-space.
With functions, as explored in one of the other slides, it can be decomposed into a sum of bases functions. Thus, the direction is just the components of the different functions present, e.g. sinusoids at different frequencies. Unfortunately, this doesn't translate into a spatial representation for direction, but it's still the same idea -- some component along each basis function.
There are a number of ways to define magnitude (norm) of a vector / function, but the most intuitive is likely just the integral. Even with vectors, while the L2 norm is common, the L1 norm has its uses as well (a la Intro to Robotics), and so the magnitude definition is context dependent, I think.
keenan
@geminish: Unlike the zero function, which is zero at every point, the symbol "1" here just means the usual, single number 1. The idea is that this number gets multiplied by every element of the vector, much like you might have, say, 7(x,y,z) = (7x,7y,7z).
We will try to make the distinction between scalars and vectors clear by making vectors bold. Though sometimes it can be hard to tell, based on the font (for instance, here "1" is in a regular typeface, but looks like it could perhaps be bold).
the 'unit funcion' in '1v=v' is function equal to 1 for all x?
If functions are vectors, how would we define the direction and magnitude? (To be pedantic).
I don't think it's necessary to define a direction and magnitude in the conventional sense for functions. If I'm not mistaken, a vector is simply an element of a vector space which only needs to satisfy the properties above, and since a function satisfies these constraints, it is a vector.
That said, think about what the direction of a standard vector represents. The values are simply coefficients of the corresponding bases vectors in a summation, e.g. (x, y) is just x * (1, 0) + y * (0, 1) with the standard bases. So, the direction is just a measure of components along each basis vector, which happens to have a direct spatial meaning when working with 2-space and 3-space.
With functions, as explored in one of the other slides, it can be decomposed into a sum of bases functions. Thus, the direction is just the components of the different functions present, e.g. sinusoids at different frequencies. Unfortunately, this doesn't translate into a spatial representation for direction, but it's still the same idea -- some component along each basis function.
There are a number of ways to define magnitude (norm) of a vector / function, but the most intuitive is likely just the integral. Even with vectors, while the L2 norm is common, the L1 norm has its uses as well (a la Intro to Robotics), and so the magnitude definition is context dependent, I think.
@geminish: Unlike the zero function, which is zero at every point, the symbol "1" here just means the usual, single number 1. The idea is that this number gets multiplied by every element of the vector, much like you might have, say, 7(x,y,z) = (7x,7y,7z).
We will try to make the distinction between scalars and vectors clear by making vectors bold. Though sometimes it can be hard to tell, based on the font (for instance, here "1" is in a regular typeface, but looks like it could perhaps be bold).