Could you please help me in the following problem:
Let be a proper subspace of an dimensional vector space , and let . Show that there is a linear functional on such that and for all .
Of course this is true. Namely, for any vector spaces with basis and vector space there exists a unique linear transformation such that where for . Indeed, just define . Clearly then this is a linear transformation which satisfies the condition and moreover it's clear that any linear transformation which satisfies that condition must look like that. Use this methodology here by noting that every linear functional is a linear transformation when is viewed as a one-dimensional vector space over itself.