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- Thread starter coco richie
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A linear transformation is a function that behaves like functions that give us lines through the origin when graphed on the Cartesian plane. In particular, those functions are all of the form f(x) = ax for some constant a.

It is easy to see that incrementing x by h increments the function by the same function of h (ie., it is like tacking on a copy of "the line from f(0) to f(h)" to x by moving the origin to x). Mathematically, this says f(x + h) = f(x) + f(h): the function distributes over addition as if it were multiplication.

Also like multiplication, the function can be factored out of a product: f(bx) = bf(x).

These two properties define a linear transformation, no matter how abstract the field of study is. It is just a very simple type of function that behaves very nicely with respect to addition and multiplication of the elements in its domain and image. From the properties above, it is easy to see that these functions have their own internal algebra.

We generalize this by noting that x need not be just a number, but can be any object for which it is reasonable to define addition and scaling by a number. This leads to linear algebra and other algebras.

In physical terms in one dimension, when you change something by a particular amount in the domain, the related image is changed by a constant directly proportional amount. Ie., this is similar to changing the unit of measurement from inches to centimeters or vice versa. It does not matter what the particular value of inches is, only the change in inches that determines the change in centimeters. A nonlinear transformation would not be able to do this; one would have to specify the start and endpoint of the inches measurement in order to translate correctly, ie., if I(x) = x^{2}. It is not true that I(x + h) = I(x) + I(h).

In more than one dimensional, a linear transformation can do more interesting things, such as rotation and shearing, where each dimension scales at a different rate than the others.

Another common generalization is generalizing all lines: f(x) = ax + b. These types of transformations are referred to as affine.

It is easy to see that incrementing x by h increments the function by the same function of h (ie., it is like tacking on a copy of "the line from f(0) to f(h)" to x by moving the origin to x). Mathematically, this says f(x + h) = f(x) + f(h): the function distributes over addition as if it were multiplication.

Also like multiplication, the function can be factored out of a product: f(bx) = bf(x).

These two properties define a linear transformation, no matter how abstract the field of study is. It is just a very simple type of function that behaves very nicely with respect to addition and multiplication of the elements in its domain and image. From the properties above, it is easy to see that these functions have their own internal algebra.

We generalize this by noting that x need not be just a number, but can be any object for which it is reasonable to define addition and scaling by a number. This leads to linear algebra and other algebras.

In physical terms in one dimension, when you change something by a particular amount in the domain, the related image is changed by a constant directly proportional amount. Ie., this is similar to changing the unit of measurement from inches to centimeters or vice versa. It does not matter what the particular value of inches is, only the change in inches that determines the change in centimeters. A nonlinear transformation would not be able to do this; one would have to specify the start and endpoint of the inches measurement in order to translate correctly, ie., if I(x) = x

In more than one dimensional, a linear transformation can do more interesting things, such as rotation and shearing, where each dimension scales at a different rate than the others.

Another common generalization is generalizing all lines: f(x) = ax + b. These types of transformations are referred to as affine.

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In fact, if f and g are linear, then f(g(x)) = g(f(x)), so we may as well write fg(x) = gf(x).

How would you word this equation? Is it f times g(x) is equal to g times f(x)?

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These two properties define a linear transformation, no matter how abstract the field of study is.

Just to be clear slider, are you referring to multiplication and addition as the two properties of the functions that define linear transformation? What other choices do we have that makes a function "non-linear"?

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AlephZero

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The image of a function is the set of all possible output values from a function.

For example, if we are talking about real numbers

For the function f(x) = sqrt(x-1),

the domain is all numbers >= 1 (because you can't take the square root of a negative umber)

The image is all numbers >= 0 (because zero or any positive number is the square root of some number)

The reason linear transformations are defined in terms of vector spaces is because a "vector space" is the simplest mathematical structure for which they CAN be defined.

If you don't know what a vector space is, just think about a specific example of a vector space, for example the points on a plane, or in 3-D space, in "ordinary" (Euclidean) geometry.

A function is just some rule that takes a point and produces another point.

If you apply a function to all the points that make up a "picture" on the plane, you will transform it into another picture. Usuall this will "distort" the picture in some way, and in particular straight lines will finish up curved.

A linear transformation is one where every possible straight line in the original picture is transformed into another straight line (i.e. never into a curved line).

There are several simple linear transformations, for example

* Sliding (translating) the whole picture left and right, or up and down

* Enlarging or reducing the picture, either by the same amount in all directions or just in one direction (e.g. horizontally but not vertically)

* Rotating the picture by any angle around any point.

* "Shear transformations" that change rectangles into a parallelograms, or in other words they can change the size of the angles between lines in the picture, but the lines themselves remain straight.

It follows from the definition of a linear transformation that the result of any sequence of linear transformations applied one after another is equivalent to just one (more complicated) linear transformation.

In fact this idea can be more useful if you work it backwards, and demonstrate that any complicated linear transformation can be broken down into a sequense of "simple" transformations like the ones I described above. This is the basic idea behind a lot of the math used to generate computer graphics.

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The image of a function, also called the range, is the set of all its outputs. The outputs are usually called the values of the function.

The image of a subset, S, of the domain is the set of all of the function's outputs, f(x), such that x is in S.

The codomain of a function is a set specified in the definition of the function which must contain all of its outputs, although it may contain other elements that aren't outputs of the function. The range is a subset of the codomain.

Notation. Let x be an element of the set D, the domain of the function f. Then f(x) is the value of f on x, and is an element of the range of f. The range of f may be denoted f(D). Let S be a subset of D. Then the image of S "under" f is f(S), a subset of the range.

For example, let f be a function from the set {1,2,3,4} (the domain) to the set {%,@,56,12,*,#} (the codomain) such that f(1) = *, f(2) = @, f(3) = @, f(4) = %. Then the image of the subset {1,2} is {*,@}. So is the image of the subset {1,2,3}. The image of the function (equivalently, the range of the function), i.e. the image of the whole domain, is {3,@,%}.

For example, let g be a function from the real numbers to the real numbers, R, such that g(x) = 2x. The domain is R. The range is R. The image of the interval (0,1), that is all real numbers strictly between 0 and 1, is the interval (0,2).

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Just to be clear slider, are you referring to multiplication and addition as the two properties of the functions that define linear transformation? What other choices do we have that makes a function "non-linear"?

No, rather linear transformations distribute over sums and scalings, similar to the way multiplication by a constant distributes over addition and scaling. That is, the linear transformation of a sum is the same as the sum of the linear transformation applied to each term, and furthermore, the linear transformation of a scaled object is the same as the scalar applied to the transformation of the unscaled object. You will find this type of transformation in other guises as you study more mathematics.

They are quite neat. For example, suppose you know that L(5) = 3 and L is a linear transformation. You then know every value of L(x) for every x! For example, L(1) = L((1/5)*5) = (1/5)L(5) = 3/5.

For n dimensions, you will need to know n independent input-output pairs in order to know every value of the transformation. We need them to be independent because, for example, it is of no use to know two output values if the input values are multiples of each other, as that gives no new information, as seen if you worked the example above. The general argument brings us to linear independence, which brings about the usefulness of basis vectors and matrices.

For example, in two dimensions, you may be operating on ordered pairs of numbers that have addition and scalar multiplication defined. This is called a vector space. If v and w are elements in this space, called vectors, and r is a real number, called a scalar because we only use numbers to scale our vectors, then we can now mean something by L(v + w) = L(v) + L(w) and L(rw) = rL(w).

Ie., suppose we have L(1, 1) = (-1, 1) and L(1, -1) = (1, 1). We can now find the result of applying this transformation to any 2-dimensional vector.

You will study these concepts in detail in a course on linear algebra.

A related concept is that of an affine transformation. A very common 1-dimensional affine transformation (at least here in the U.S.A.) is the affine transformation between degrees Celsius and degrees Fahrenheit. Knowing that it is affine, and knowing two specific points, that 0 degrees Celsius is 32 degrees Fahrenheit (freezing point of water at STP) and 100 degrees Celsius is 212 degrees Fahrenheit (boiling point of water at STP) allows us to easily derive the full relationship for any particular temperature.

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A translation is *not* a linear transformation; it is an affine transformation.

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How would you word this equation? Is it f times g(x) is equal to g times f(x)?

I would ignore this aspect until you have studied linear algebra further.

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Map and mapping mean the same as function: a pair of sets, called the domain and codomain of the function, and a rule associating each element of the domain (input) with no more than one element of the codomain (output) each.

A vector space is any mathematical structure which obeys the http://cnx.org/content/m10419/latest/ [Broken].

Transformation also means function, but the word is typically applied to a function whose inputs and outputs are vectors, or points of an affine space, as the case may be.

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A translation is *not* a linear transformation; it is an affine transformation.

I am well aware of this which is why I posted the question.

Coco, you asked for a plain English explanation, rather than the mathematical discussion presented so far.

Well here goes.

A linear transformation is a formal mathematical way of presenting the idea of proportionality or the english "is proportional to".

For example

If I can shovel 1/2 a ton of sand in 1 hour I can expect to shovel 1 ton in 2 hours. That is proportionality.

The transformation is linear.

But consider my electricity bill

If I pay $100 for 100 units of electricity can I expect to pay $200 for 200 units?

Well, where I live at least, the answer is no because some of that first $100 is standing charge, which is the same, no matter how much electricity I use.

So I actually pay $50 standing charge and 50c per unit so I pay $150 for 200 units.

That is a non linear transformation.

All the rest is just fancy names for this.

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In fact, if f and g are linear, then f(g(x)) = g(f(x)), so we may as well write fg(x) = gf(x).

Possible interpretation: composition of linear functions is commutative. Counter example: rotations in

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Possible interpretation: composition of linear functions is commutative. Counter example: rotations inR^{3}. So I guess it's not that... Could you disambiguate?

Indeed, I'm going to remove that statement; it only holds in 1 dimension and specal cases and is not very enlightening.

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