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Think of the columns of the matrix A as basis vectors of a coordinate system represented in global coordinates, and the vector v as the components of a vector in that coordinate system. Then the product A * v transforms the vector v into the global coordinate system.
Carrying this forward, matrix multiplication A * B gives the combined representation of two coordinate transformations.
Hmm. As someone who has never understood/visualized/been taught matrix multiplication in any other way than how it's shown on the linked page, could someone explain the alternative? I.e., if this is new, what mental model do you currently have? Presumably that one would be new and useful for me. Edit: "Typically this visualization isn't taught" -- that's what I"m asking. What visualization were you taught, if not thi…
Typically, this visualization isn't taught, in my experience. What's taught is the formula for computing each cell of the result matrix (cell i,j being the dot product of row i of the first matrix with column j of the second). While this is, of course correct, and also the most efficient way to compute the multiplication manually, it's not always clear why the formula is correct. Hopefully the visualization on the li…
Here's how you can apply this interpretation of matrix multiplication: Think of the columns of the matrix A as basis vectors of a coordinate system represented in global coordinates, and the vector v as the components of a vector in that coordinate system. Then the product A * v transforms the vector v into the global coordinate system. Carrying this forward, matrix multiplication A * B gives the combined representat…
Here's how you can apply this interpretation of matrix multiplication: Think of the columns of the matrix A as basis vectors of a coordinate system represented in global coordinates, and the vector v as the components of a vector in that coordinate system. Then the product A * v transforms the vector v into the global coordinate system. Carrying this forward, matrix multiplication A * B gives the combined representat…
The first type is a change of basis where the columns of the matrix are the old basis represented in the new basis.
The second type is a linear function that takes in a vector and outputs another vector.
(If the mafiosi could read the following paper before rushing to click down-vote, then I'd be much obliged. http://www.math.rutgers.edu/~zeilberg/mamarimY/DM85.pdf)
If you want more, I can definitely recommend the video lectures from MIT's linear algebra course by Prof Gilbert Strang: http://web.mit.edu/18.06/www/videos.shtml
AB = C,
C has the same number of rows as A and the same number of columns as B, and each column j of C is from column j of B acting as coefficients in a linear combination of all the columns of A.
Next: For the set of real numbers R, positive integers m and n, m by n real matrix A, real numbers a and b, and n by 1 x and y, we have that
A(ax + by) = (aA)x + (bA)y
so that A is linear.
If we regard x and y as vectors in the n-dimensional real vector space R^n, then we have that A is a function
A: R^n --> R^m
and a linear operator which can be good to know.
Then the subset of R^n
K = { x | x in R^n and Ax = 0 }
(where 0 is the m by 1 matrix of all zeros) is important to understand. E.g., K = [0] if and only if function A is 1-1. If m = n, then A has an inverse A^(-1) if and only if A is 1-1.
With C = AB, the matrix product is the same as function composition so that
Cx = (AB)x = A(Bx)
which also can be good to know and, of course, uses just the associative law of matrix multiplication.
That matrix multiplication is associative is a biggie -- sometimes says some big things, e.g., is the core of duality theory in linear programming.
And the situation is entirely similar for the complex numbers C in place of the real numbers R.
If you want more, I can definitely recommend the video lectures from MIT's linear algebra course by Prof Gilbert Strang: http://web.mit.edu/18.06/www/videos.shtml
This tutorial is also pretty good for geometrical intuition: http://www.cns.nyu.edu/~eero/NOTES/geomLinAlg.pdf
This article was posted a few weeks ago, helped me understand matrix operations I've been trying to wrap my head around for years: http://betterexplained.com/articles/linear-algebra-guide/