Let Z be a proper subspace of an n-dimensional vector space X

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March 29th 2011, 03:24 AM
raed

Let Z be a proper subspace of an n-dimensional vector space X

Dear Colleagues,

Could you please help me in the following problem:
Let $Z$ be a proper subspace of an $n-$ dimensional vector space $X$ , and let $x_{0}\in X-Z$ . Show that there is a linear functional $f$ on $X$ such that $f(x_{0})=1$ and $f(x)=0$ for all $x\in Z$ .

Regards,

Raed.
March 29th 2011, 06:25 AM
Drexel28

Quote:

Originally Posted by raed

Dear Colleagues,

Could you please help me in the following problem:
Let $Z$ be a proper subspace of an $n-$ dimensional vector space $X$ , and let $x_{0}\in X-Z$ . Show that there is a linear functional $f$ on $X$ such that $f(x_{0})=1$ and $f(x)=0$ for all $x\in Z$ .

Regards,

Raed.

Of course this is true. Namely, for any vector spaces $V$ with basis $\{x_1,\cdots,x_n\}$ and vector space $W$ there exists a unique linear transformation $T$ such that $T(x_k)=w_k$ where $w_k\in W$ for $k=1,\cdots,n$ . Indeed, just define $\displaystyle T\left(\sum_{r=1}^{n}\alpha_r x_r\right)=\sum_{r=1}^{n}\alpha_r w_r$ . 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 $\varphi:V\to F$ is a linear transformation when $F$ is viewed as a one-dimensional vector space over itself.
March 29th 2011, 07:35 AM
raed

Thank you very much for your reply.
March 29th 2011, 11:56 AM
Drexel28

Quote:

Originally Posted by raed

Thank you very much for your reply.

Just as a point, you should press the 'thank button' located on the bottom left of the posts for which you appreciate.