- #1
mahler1
- 222
- 0
The problem statement, all variable
Let ##\phi_1,...,\phi_n \in V^*## all different from the zero functional. Prove that
##\{\phi_1,...,\phi_n\}## is basis of ##V^*## if and only if ##\bigcap_{i=1}^n Nu(\phi_i)={0}##.
The attempt at a solution.
For ##→##: Let ##\{v_1,...,v_n\}## be a basis of ##V## such that ##\phi_i(v_j)=δ_{ij}##. Now let ##x \in \bigcap_{i=1}^n Nu(\phi_i)={0}##, as ##x \in V## then ##x=α_1v_1+...+α_nv_n##.
By hypothesis ##\phi_i(x)=0## for all ##1\leq i \leq n##. But this means ##0=\phi_i(x)=\phi_i(α_1v_1+...+α_nv_n)=α_i## for all ##1\leq i \leq n##, it follows ##x=0##.
I don't know how to show the other implication. For ←, in the exercise is given the suggestion: if ##\{ψ_1,...,ψ_n\}## is a basis of ##V^*##, one can try to show that the matrix obtained from writing ##\phi_1,...,\phi_n## in coordinates with respect to the basis ##\{ψ_1,...,ψ_n\}##, is an invertible matrix, I am stuck and I haven't got a clue on how to use this idea, I would like any hints or another idea to prove the remaining implication.
Let ##\phi_1,...,\phi_n \in V^*## all different from the zero functional. Prove that
##\{\phi_1,...,\phi_n\}## is basis of ##V^*## if and only if ##\bigcap_{i=1}^n Nu(\phi_i)={0}##.
The attempt at a solution.
For ##→##: Let ##\{v_1,...,v_n\}## be a basis of ##V## such that ##\phi_i(v_j)=δ_{ij}##. Now let ##x \in \bigcap_{i=1}^n Nu(\phi_i)={0}##, as ##x \in V## then ##x=α_1v_1+...+α_nv_n##.
By hypothesis ##\phi_i(x)=0## for all ##1\leq i \leq n##. But this means ##0=\phi_i(x)=\phi_i(α_1v_1+...+α_nv_n)=α_i## for all ##1\leq i \leq n##, it follows ##x=0##.
I don't know how to show the other implication. For ←, in the exercise is given the suggestion: if ##\{ψ_1,...,ψ_n\}## is a basis of ##V^*##, one can try to show that the matrix obtained from writing ##\phi_1,...,\phi_n## in coordinates with respect to the basis ##\{ψ_1,...,ψ_n\}##, is an invertible matrix, I am stuck and I haven't got a clue on how to use this idea, I would like any hints or another idea to prove the remaining implication.