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I only recently realized that multiplying a matrix with a column vector is equivalent to a linear combination of the vector's columns. This explains the importa

by samus 4y ago

I only recently realized that multiplying a matrix with a column vector is equivalent to a linear combination of the vector's columns. This explains the importance of matrix multiplication since 95% of Linear Algebra is calculating linear combinations of vectors.

curriculum 4y ago
Indeed, and this leads to another important interpretation: matrix multiplication is function evaluation. Arbitrary functions which take in vectors and output vectors can be very complex and thus difficult to reason about. A useful simplifying assumption is that of linearity: that f applied to a linear combination is just a linear combination of f applied to each piece of the combination separately. Linear algebra, broadly speaking, is the study of functions of this kind and the properties that emerge from making the linearity assumption. It turns out that, if we assume a function f is linear, all of the information about that function is contained in what it does to a set of basis vectors. We can in essence "encode" the function by a table of numbers (a matrix), where the kth column contains the result of f applied to the kth basis vector. In this way, given a basis, any linear transformation f has a matrix A which compactly represents it. Since f is linear, to compute f(v) I could write v in my chosen basis then apply f to each basis vector and recombine. Alternatively, I could write the matrix A representing f in that basis, and then multiply Av. The two are equivalent: that is, Av = f(v). And so matrix-vector multiplication is "just" evaluating the function f.
Koshkin 4y ago
> vector's columns Matrix columns, you mean. I think it is Strang who is a staunch proponent of teaching this to students early on. It is indeed helpful to realize that the matrix can be seen as a collection of the images of the basis vectors under the linear transformation the matrix represents.
- samus 4y ago
  Thanks for the correction. Too late to fix it though. Yes, teaching it like this would instantly eliminate all the mystery around it.

by samus 4y ago

curriculum 4y ago

Indeed, and this leads to another important interpretation: matrix multiplication is function evaluation. Arbitrary functions which take in vectors and output vectors can be very complex and thus difficult to reason about. A useful simplifying assumption is that of linearity: that f applied to a linear combination is just a linear combination of f applied to each piece of the combination separately. Linear algebra, broadly speaking, is the study of functions of this kind and the properties that emerge from making the linearity assumption. It turns out that, if we assume a function f is linear, all of the information about that function is contained in what it does to a set of basis vectors. We can in essence "encode" the function by a table of numbers (a matrix), where the kth column contains the result of f applied to the kth basis vector. In this way, given a basis, any linear transformation f has a matrix A which compactly represents it. Since f is linear, to compute f(v) I could write v in my chosen basis then apply f to each basis vector and recombine. Alternatively, I could write the matrix A representing f in that basis, and then multiply Av. The two are equivalent: that is, Av = f(v). And so matrix-vector multiplication is "just" evaluating the function f.

Koshkin 4y ago

> vector's columns Matrix columns, you mean. I think it is Strang who is a staunch proponent of teaching this to students early on. It is indeed helpful to realize that the matrix can be seen as a collection of the images of the basis vectors under the linear transformation the matrix represents.

samus 4y ago
Thanks for the correction. Too late to fix it though. Yes, teaching it like this would instantly eliminate all the mystery around it.