any particular solution plus the general solution to the homogeneous equation.(adsbygoogle = window.adsbygoogle || []).push({});

I'm having difficuilty understanding this proof from my lecture notes

Theorem

: Let T : V → W be a linear transformation. Let w ∈ W and suppose T(u0) = w

T(v) = 0. where v ∈ V (the kernel )

to prove:

T(u) = w, where u = u0 + v

proof: (from my lecture notes)

First, we show that every vector of the form u = u0 + v

where T(u0) = w and T(v) = 0, satisfies T(u) = w:

T(u) = T(u0 + v) = T(u0) + T(v) = w + 0 = w

Now we show that every solution looks like this.

Suppose that T(u) = w. Let v = u − u0.

Then u0 + v = u0 + (u − u0) = u and v is in the kernel of

T: T(v) = T(u − u0) = T(u) − T(u0) = w − w = 0

I can follow the first part of the proof. The second part is where I'm having difficulty. I'm not sure how the last three lines show that every solution looks like u = u0 + w, particularly the second last line. Is that not showing what we were already given?

Any help would be appreciated

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# I Proof that the general solution of a linear equation is....

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