Matrix rep. of Linear Transformation

  • Thread starter eckiller
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  • #1
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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?
 

Answers and Replies

  • #2
HallsofIvy
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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
 

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