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So, matrices are a convenient way to write down linear transformations. But then the student might ask, why study linear transformations at all? Just because th

So, matrices are a convenient way to write down linear transformations. But then the student might ask, why study linear transformations at all? Just because they have nice properties? Why this particular set of nice properties, and not some other set? As a rule of thumb, a good math explanation shouldn't start with axioms and claim that they are "nice". First give some intuitive examples, and only then say which axioms they satisfy. For linear transformations, one possible avenue is start with the notion of derivative. If we take a real-valued function, its derivative at a particular point is just a single number, which represents the function's rate of change. But what if we have a function that accepts, say, two real numbers and outputs three? It turns out that the natural generalization of "derivative" to such functions is a rectangular array of numbers: dy1/dx1 dy1/dx2 dy2/dx1 dy2/dx2 dy3/dx1 dy3/dx2 If we know these numbers (and nothing else), we can linearly approximate the values of a function near a particular point, with at most quadratic error. Now let's say we have two functions. The first one takes two numbers and outputs three, and the second one takes three numbers and outputs four. If we compose them together, can we find the derivative of the composite from the two simpler derivatives by some kind of chain rule, like the one we have for ordinary real-valued functions? It turns out that yes, we can, if we replace the product of two numbers with the product of two matrices (defined in a particular way). Now it's easy to explain what linear transformations are. They are just multidimensional functions whose derivative (matrix) is the same at every point. They are just like one-dimensional linear functions, whose derivative (number) is the same at every point. (For convenience, people also say that every linear transformation must take the point (0,0,...) to the point (0,0,...), so that matrices correspond one-to-one to linear transformations and vice versa.) If you want to work with linear transformations effortlessly, there's a lot more intuition to develop, but this should serve for the basics.

j2kun 11y ago
In most curricula (in the US at least) matrix multiplication is taught before calculus. So that sucks.
- tomadi 11y ago
  Why would you want to teach "every well behaved function has a family of linear functions associated with it" before you teach how to work with linear functions?
  - j2kun 11y ago
    Who said anything about linear functions? Matrix multiplication is usually taught in high school as a standalone topic for no apparent reason.
    - gohrt 11y ago
      Not in the USA, it isn't. http://www.corestandards.org/Math/Content/HSN/VM/ http://www.corestandards.org/Math/Content/HSN/VM/ CCSS.MATH.CONTENT.HSN.VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. CCSS.MATH.CONTENT.HSN.VM.C.12 (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area. Math education gets a bad rap in the USA, because anything that students don't remember learning, is claimed to have never have been taught. And that is just not true. Math teachers aren't as dumb as people pretend they are.
      - j2kun 11y ago
        The word linear is never used in that part of the standard, nor is any relationship to calculus expressed explicitly (and there is not really any relationship to anything else, just saying that a matrix can be used to express incidences in a network is useless if they never study networks). Probably because calculus is not part of the CC standard, which is my point exactly. Besides, Common Core is not what most living people in the US experienced in school. I have seen the standards for high school math teachers and trust me, they are low. Most of my classmates in undergrad (which was supposed to be an excellent program for math teachers) are now high school teachers. Their complaints about how much they hate basic linear algebra and "just want to be done with it so they can go teach high school" are still ringing in my ears.
tanderson92 11y ago
> If we know these numbers (and nothing else), we can linearly approximate the values of a function near a particular point, with at most quadratic error. Ignoring the fact that you need the function's values actually at the particular point in question, you do in fact need to know something else: you need to know that the function in question has sufficient regularity (enough smoothness) to allow an application of Taylor's theorem.
- cousin_it 11y ago
  Yeah, I'm glossing over a bunch of things.
east2west 11y ago
Matrix is not a convenient way to write linear transformation; it is a good way to reason about linear transformation, because among other things matrix product is function composition. Loose definition and incoherent reasoning are not good motivation for anything let alone a mathematical subject. Bringing matrix analysis as an introduction to linear algebra. For example, there are different ways of writing chain rule for matrices with radically different looking formulas; higher derivatives of vector functions are multilinear forms; convergence in multidimensinal space has too many subtleties for beginning students. Solving systems of linear equation is a perfect way of introducing linear algebra. Not only is it vitally important in math and used everywhere, it is also almost half of linear algebra. Gaussian elimination is the elementary school method formulated as an algorithm and, not coincidentally a sequence of linear transformations culminating in LU decomposition because we can represent a sequence of linear transformation as a series of matrix products. Writing Gaussian elimination in partitioned matrix form, which includes regular scalar manipulation as a special case, results in Schur's complement, and taking inverse of that is a good way to deductively prove the matrix inversion lemma, an important formula in numerical analysis and signal processing and usually taught in advanced courses. Nothing I mentioned is inaccessible to an undergraduate. Linear algebra is an unique mathematical subject because it doesn't require much basics to begin and is very visual, unlike calculus which needs a good foundation in analytic geometry and has many techniques what beginning students call tricks, not to mention mental obstacles like limits. Mathematical intuition comes with practice and experience and mathematical maturity, not dumbing down the topic. I agree motivations should be taught along with any mathematical subjects, but they should be mathematical motivations. Applications are important too, but teaching well and thoroughly theory and application at the same time is impossible and undesirable; different people need different applications which in turn emphasize different aspects of theory; it is just too much.

j2kun 11y ago

In most curricula (in the US at least) matrix multiplication is taught before calculus. So that sucks.

tomadi 11y ago
Why would you want to teach "every well behaved function has a family of linear functions associated with it" before you teach how to work with linear functions?
- j2kun 11y ago
  Who said anything about linear functions? Matrix multiplication is usually taught in high school as a standalone topic for no apparent reason.
  - gohrt 11y ago
    Not in the USA, it isn't. http://www.corestandards.org/Math/Content/HSN/VM/ http://www.corestandards.org/Math/Content/HSN/VM/ CCSS.MATH.CONTENT.HSN.VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. CCSS.MATH.CONTENT.HSN.VM.C.12 (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area. Math education gets a bad rap in the USA, because anything that students don't remember learning, is claimed to have never have been taught. And that is just not true. Math teachers aren't as dumb as people pretend they are.
    - j2kun 11y ago
      The word linear is never used in that part of the standard, nor is any relationship to calculus expressed explicitly (and there is not really any relationship to anything else, just saying that a matrix can be used to express incidences in a network is useless if they never study networks). Probably because calculus is not part of the CC standard, which is my point exactly. Besides, Common Core is not what most living people in the US experienced in school. I have seen the standards for high school math teachers and trust me, they are low. Most of my classmates in undergrad (which was supposed to be an excellent program for math teachers) are now high school teachers. Their complaints about how much they hate basic linear algebra and "just want to be done with it so they can go teach high school" are still ringing in my ears.

tanderson92 11y ago

> If we know these numbers (and nothing else), we can linearly approximate the values of a function near a particular point, with at most quadratic error. Ignoring the fact that you need the function's values actually at the particular point in question, you do in fact need to know something else: you need to know that the function in question has sufficient regularity (enough smoothness) to allow an application of Taylor's theorem.

cousin_it 11y ago
Yeah, I'm glossing over a bunch of things.

east2west 11y ago

Matrix is not a convenient way to write linear transformation; it is a good way to reason about linear transformation, because among other things matrix product is function composition. Loose definition and incoherent reasoning are not good motivation for anything let alone a mathematical subject. Bringing matrix analysis as an introduction to linear algebra. For example, there are different ways of writing chain rule for matrices with radically different looking formulas; higher derivatives of vector functions are multilinear forms; convergence in multidimensinal space has too many subtleties for beginning students. Solving systems of linear equation is a perfect way of introducing linear algebra. Not only is it vitally important in math and used everywhere, it is also almost half of linear algebra. Gaussian elimination is the elementary school method formulated as an algorithm and, not coincidentally a sequence of linear transformations culminating in LU decomposition because we can represent a sequence of linear transformation as a series of matrix products. Writing Gaussian elimination in partitioned matrix form, which includes regular scalar manipulation as a special case, results in Schur's complement, and taking inverse of that is a good way to deductively prove the matrix inversion lemma, an important formula in numerical analysis and signal processing and usually taught in advanced courses. Nothing I mentioned is inaccessible to an undergraduate. Linear algebra is an unique mathematical subject because it doesn't require much basics to begin and is very visual, unlike calculus which needs a good foundation in analytic geometry and has many techniques what beginning students call tricks, not to mention mental obstacles like limits. Mathematical intuition comes with practice and experience and mathematical maturity, not dumbing down the topic. I agree motivations should be taught along with any mathematical subjects, but they should be mathematical motivations. Applications are important too, but teaching well and thoroughly theory and application at the same time is impossible and undesirable; different people need different applications which in turn emphasize different aspects of theory; it is just too much.