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mrpace

Member

Registered: 2012-08-16

Posts: 88

Linear algebra proof

Let T : V --> W be a linear transformation with ker(T) = {0}

Suppose u1, u2 ∈ V are linearly independent.

Prove that T(u1) and T(u2) are linearly independent vectors in W.

Just not sure how to construct this. Any help would be great, thanks.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Linear algebra proof

hi mrpace

It's been a long time since I did any linear algebra, so this may not work but:

I think the kernel property implies T has an inverse, say S  (Cannot find an on-line reference that confirms this.)

So assume T(u1) = k.T(u2), apply the inverse matrix S, and hence reach a contradiction.

LATER EDIT>  Done it myself thus:

Let's assume two vectors map onto the same vector:

ie.  T(u1) = T(u2) = v

then T(u1) - T(u2) = v - v = 0

But T is linear so

T(u1 - u2) = 0

But the kernel of T is just the zero vector so

u1 - u2 = 0  => u1 = u2

So every vector maps onto a unique vector => T will have an inverse, S.

Bob

Last edited by Bob (2014-08-05 20:16:54)

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