A Question in Linear Tranformations

[beanz]

hi

I am a college student who has just started doing Linear Algebra. This is not a homework question, just something abot Linear Transformations that i don't understand. I hope someone can help me. Here it goes:

Consider Vector spaces V and W (over R^4) and a matrix 'A' which maps an element from V to W.

1. Why is it that the basis for the column space for A is exactly the basis for the range space of V in W? [i.e why is dim(col(A))=dim (range(V))]

2. Why is dim (V) = dim (Null space of V) + dim (range space of V in W)?

Thanx in advance :)

 

[HallsofIvy]

The basis for V is, of course, (1, 0, 0, 0), (0, 1, 0,0), (0, 0, 1, 0), (0, 0, 0, 1). Write those as columns and multiply each by A. Do you see that multiplying A times (1, 0, 0, 0), you get exactly the first column of A? And that multiplying A time (0, 1, 0, 0) gives exactly the second column of A? Write any vector in V as a linear combination of (1, 0, 0, 0), etc. and multiplying by A gives the result, in W, as a linear combination of the columns of A.

 

[beanz]

Hahaha, Thanks. It is so obvious now that you mention it.