linalg - product of "map matrix" and "vector matrix"

[sh3rlock14]

Slides "03-linalg", 71/72

I'm not getting why the column-matrix storing the linear combination coefficients is noted as Tvj, since we are not expressing the basis vectors of W in the formula.

Also in the lecture is said that "what we get is the matrix representation of Tu" (probably meant Tv), but again: if it were so, then shouldn't it be: T1,jw1 + ... + Tm,jwm ?

Thanks in advance for the help!

[erodola]

I went through the slides and the video recording to double-check, let me now dispel your doubts:

• The column-matrix with the numbers T_1,j ... T_m,j is by definition a representation with respect to the basis w_1 ... w_m. Recall that every time we write numbers in a matrix, we do it after fixing a basis in which those numbers make sense. Since the right hand side of the equation expresses a vector in the target vector space W, the basis that underlies any matrix representation on the right hand side is the basis w_1 ... w_m.

• Why do we call it Tvj? Because that's precisely what it encodes. Do the following thought experiment: in the matrix equation of slide 71/72, replace the c vector with an indicator vector (0 0 .. 1 .. 0), where 1 is in position j. The result of the matrix-vector product in this case will yield precisely the j-th column of T, which is Tvj expressed in the basis w_1 ... w_m. If this works for you, then you can replace the indicator vector with a weighted indicator vector (0 0 .. c_j .. 0), where c_j is a scalar value in position j. Then, the matrix-vector product yields the j-th column of T, weighted by the scalar c_j. If this also works for you, then you can replace the weighted indicator vector with an entire vector of weights (c_1 c_2 .. c_j .. c_n); the matrix-vector product in this case will be the weighted sum of the columns of T, which is what is written in the slide.

• "what we get is the matrix representation of Tu": indeed, I meant Tv instead of Tu.

I hope this clarifies. If not, feel free to cast your doubts here!

Addendum: this should also clarify our previous statement that matrix representations only need to know how to map basis vectors Tvj, since for arbitrary vectors c we only need to take a linear combination of the Tvj with the coefficients contained in c.