Matrix rep. of Linear Transformation

[eckiller]

Hello all,

I am trying to understand the matrix representation of a linear transformation.

So here is my thought process.

Let B = (b1, b2, ..., bn) be a basis for V, and let Y = (y1, y2, ..., ym) be a basis for W.

T: V --> W

Pick and v in V and express as a linear combo of the basis vectors:

v = sum( ai bi, 1, n)

T(v) = sum( ai T(bi), 1, n)

i.e., the transformed vector T(v) is determined by a linear combination of the transformed basis vectors.

Now coordanitize everything relative to Y, which we can always do since it is an isomorphism.

[T(v)]_Y = sum( ai [T(bi)]_Y, 1, n)

Then we can write this linear combination as a matrix multiplication, i.e., the vectors [T(bi)]_Y give the column vectors of the matrix representation.

Anyway, it took me awhile to get this and I still doubt myself. Is my reasoning correct?

 

[HallsofIvy]

It is simpler to look at the individual basis vectors. If you have bi in a specific order, then b1 itself is represented by the ntuple (1, 0, 0,..., 0). Writing that as a column vector and multiplying it by the matrix representing T, you see that each number in the first column is multiplied by 1 and all other numbers by 0. That is, the first column is precisely the coefficients of T(b1).

Now look at b2, etc. to get the other columns

 

[M98Ranger]

Hi there,

Your thought process for understanding the matrix representation of a linear transformation is correct. By choosing a basis for both the vector spaces V and W, we can express any vector in V as a linear combination of the basis vectors and the transformed vector T(v) can also be expressed as a linear combination of the transformed basis vectors. This allows us to write T(v) as a matrix multiplication, where the column vectors of the matrix are the transformed basis vectors.

In simpler terms, the matrix representation of a linear transformation is a way of representing the transformation as a matrix, where the columns of the matrix are the transformed basis vectors. This matrix can then be used to perform computations and transformations on vectors in V.

I hope this helps clarify your understanding. Keep up the good work!