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Find a basis and dimension of a vector space

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1. Homework Statement
Find basis and dimension of [itex]V,W,V\cap W,V+W[/itex] where [itex]V=\{p\in\mathbb{R_4}(x):p^{'}(0) \wedge p(1)=p(0)=p(-1)\},W=\{p\in\mathbb{R_4}(x):p(1)=0\}[/itex]

2. Homework Equations
-Vector spaces

3. The Attempt at a Solution
Could someone give a hint how to get general representation of a vector in [itex]V[/itex] and [itex]W[/itex]?
 

Simon Bridge

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Homework Helper
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Look in your course notes when it talks about "general representation of a vector".
You have definitions of the vector spaces - so start with what the symbols turn into in English.
 
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I'm not sure what the operator ^ means in the definition of ##V## so I can't help.
For ##W##, you can notice that ##W = (X-1) \mathbb{R}_3[X] ## is isomorphic to ##\mathbb{R}_3[X]##, so what can you say about the dimension of 2 isomorphic vector spaces ? Secondly, a vector space isomorphism sends a base to a base.
 

HallsofIvy

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The only thing I have a question about is "p'(0)^p(1)= p(0)= p(-1)". The "^" typically means "and" so one condition is that p(1)= p(0)= p(-1) but the "p'(0)" is incomplete- nothing is said about the derivative at 0. What must be true about it? p in [itex]R^4[/itex] is of the form [itex]ax^4+ bx^3+ cx^2+ dx+ e[/itex] and "p(1)= p(0)= p(-1)" requires that [itex]a+ b+ c+ d+ e= e= a- b+ c- d+ e[/itex]. But, again, what is required of "p'(0)= d"?
 

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