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cartonn30gel
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This is not really a homework problem but it relates to a number of problems, so I thought this would be the most appropriate place to post it.
The basic question is about how we define linear dependence in a vector space. For a vector space over some field [tex]\mathbb{F}[/tex], we know that the vectors [tex]v_1,v_2,...,v_n[/tex] are linearly independent if the only solution to [tex] a_1v_1+a_2v_2+...+a_nv_n= 0[/tex] is [tex]a_1=a_2=...=a_n=0[/tex]. And if some list of vectors are not linearly independent, they are linearly dependent. This means if we can find constants [tex]b_1,..., b_n[/tex] that are NOT ALL ZERO where [tex]b_1v_1+b_2v_2+...+b_nv_n=0[/tex], these vectors are linearly dependent.
Now when we think about the polynomial space (let's say over some field) [tex]P_m(\mathbb{F})[/tex], we need to reconsider these definitions right? I have never seen a different defintion of linear independence for the polynomial space but I'm assuming it would be like this:
The vectors [tex]p_1(z),...,p_m(z)[/tex] are linearly independent if the only solution to [tex]c_1p_1(z)+...+c_mp_m(z)=0[/tex] FOR ALL z is [tex]c_1=...=c_m=0[/tex].
And linear dependence would be this: If we can find constants [tex]c_1, ..., c_m[/tex] that are not all zero FOR SOME z where [tex]c_1p_1(z)+...+c_mp_m(z)=0[/tex], these vectors are linearly dependent.
Notice that if linear dependence on the polynomial space is defined this way, it is actually the exact negation of linear independence in the polynomial space. But if linear dependence is defined in the similar way but FOR ALL z, then it is NOT the exact negation of linear independence in the polynomial space. And then, we start encountering vectors that are neither linearly independent nor linearly dependent.
Please try to shed some light on this. Is my given definitions for linear independence and dependence for the polynomial space correct?
Known definitions of linear independence and dependence are given above.
There isn't really a solution. It is just a question about definitions.
Homework Statement
The basic question is about how we define linear dependence in a vector space. For a vector space over some field [tex]\mathbb{F}[/tex], we know that the vectors [tex]v_1,v_2,...,v_n[/tex] are linearly independent if the only solution to [tex] a_1v_1+a_2v_2+...+a_nv_n= 0[/tex] is [tex]a_1=a_2=...=a_n=0[/tex]. And if some list of vectors are not linearly independent, they are linearly dependent. This means if we can find constants [tex]b_1,..., b_n[/tex] that are NOT ALL ZERO where [tex]b_1v_1+b_2v_2+...+b_nv_n=0[/tex], these vectors are linearly dependent.
Now when we think about the polynomial space (let's say over some field) [tex]P_m(\mathbb{F})[/tex], we need to reconsider these definitions right? I have never seen a different defintion of linear independence for the polynomial space but I'm assuming it would be like this:
The vectors [tex]p_1(z),...,p_m(z)[/tex] are linearly independent if the only solution to [tex]c_1p_1(z)+...+c_mp_m(z)=0[/tex] FOR ALL z is [tex]c_1=...=c_m=0[/tex].
And linear dependence would be this: If we can find constants [tex]c_1, ..., c_m[/tex] that are not all zero FOR SOME z where [tex]c_1p_1(z)+...+c_mp_m(z)=0[/tex], these vectors are linearly dependent.
Notice that if linear dependence on the polynomial space is defined this way, it is actually the exact negation of linear independence in the polynomial space. But if linear dependence is defined in the similar way but FOR ALL z, then it is NOT the exact negation of linear independence in the polynomial space. And then, we start encountering vectors that are neither linearly independent nor linearly dependent.
Please try to shed some light on this. Is my given definitions for linear independence and dependence for the polynomial space correct?
Homework Equations
Known definitions of linear independence and dependence are given above.
The Attempt at a Solution
There isn't really a solution. It is just a question about definitions.
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