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nilwill
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Hi there. I'm a long time reader, first time poster. I'm an undergraduate in Math and Economics and I am having trouble in Linear Algebra. This is the first class I have had that focuses solely on proofs, so I am in new territory.
note Although the question doesn't state it, I think P is supposed to be a projection.
Let W be a vector space. Let P:W→W be a linear map s.t. P2=P.
Show that W= KerP + ImP and KerP[itex]\cap[/itex]ImP
namely, W is the direct sum of KerP and ImP
Hint: To show W is the sum, write an element of W in the form w=w-P(w)+P(w)
I am unsure of any relevant equations.
KerP={z|P(z)=0}
ImP={P(v)|v[itex]\in[/itex]W}
I am kind of fuzzy on the meaning of P2=P, and this is where I am stuck.
Would an example be something like:
P=[itex]\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right)[/itex]P2=[itex]\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) \times \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right)[/itex] =P?
From this example: for KerP=P(z)=0 , we would need to satisfy z+0=0, so z=0
For some v[itex]\in[/itex]W, the ImP is P(v)=v.
KerP[itex]\cap[/itex]ImP=0→ P(v)=P(z) iff v=0 so z=v where the two intersect.I am unsure where to go from here, or if I'm even doing this correctly. If someone could nudge me along towards and answer without giving the me the full proof, it would be much appreciated. I've been stuck on this problem for 4 hours over two days and I can't seem to figure it out.
Homework Statement
note Although the question doesn't state it, I think P is supposed to be a projection.
Let W be a vector space. Let P:W→W be a linear map s.t. P2=P.
Show that W= KerP + ImP and KerP[itex]\cap[/itex]ImP
namely, W is the direct sum of KerP and ImP
Hint: To show W is the sum, write an element of W in the form w=w-P(w)+P(w)
Homework Equations
I am unsure of any relevant equations.
The Attempt at a Solution
KerP={z|P(z)=0}
ImP={P(v)|v[itex]\in[/itex]W}
I am kind of fuzzy on the meaning of P2=P, and this is where I am stuck.
Would an example be something like:
P=[itex]\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right)[/itex]P2=[itex]\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) \times \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right)[/itex] =P?
From this example: for KerP=P(z)=0 , we would need to satisfy z+0=0, so z=0
For some v[itex]\in[/itex]W, the ImP is P(v)=v.
KerP[itex]\cap[/itex]ImP=0→ P(v)=P(z) iff v=0 so z=v where the two intersect.I am unsure where to go from here, or if I'm even doing this correctly. If someone could nudge me along towards and answer without giving the me the full proof, it would be much appreciated. I've been stuck on this problem for 4 hours over two days and I can't seem to figure it out.