wont we need to somehow reduce the length of the vector also? Because the original vector has the length of (x^2 + y^2)^0.5 but the new projected vector will have a length of only (x) ?
Am i missing something or do we need to do it when we are doing similar transformations in R^3 ? for example when we are trying to project a vector in R^3 onto the x-y plane then what will be the matrix like? will be multiply the transformation matrix by
(x^2+y^2)^0.5 / (x^2+y^2 +z^2)^0.5
OR
(x^2+y^2 +z^2)^0.5 /(x^2+y^2)^0.5
?
Another way to do this:
(I assume you mean the matrix relative to the standard basis <1, 0> and <0, 1>. A linear transformation only has a basis relative to some basis.)
A vector in the direction of the line x= -2y is v= -2i+ j. The projection of u= i onto that is given by which, here, is .
The projection of w= j onto that is given by .
Since applying a linear transformation to the basis vectors in turn gives the columns of the matrix, the matrix corresponding to this projection, in the standard basis, is