Hello all,(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Matrix rep. of Linear Transformation

**Physics Forums | Science Articles, Homework Help, Discussion**