Please can you guys help me to solve the following questions
Q. Let Z be a proper subspace of an n-dimensional vector space X, and let x_0 \in X-Z. Show tha there is an linear functional f on X such that f(x_0)=1 and f(x)=0 for all x\in Z
Please can you guys help me to solve the following questions
Q. Let Z be a proper subspace of an n-dimensional vector space X, and let x_0 \in X-Z. Show tha there is an linear functional f on X such that f(x_0)=1 and f(x)=0 for all x\in Z
You may be overthinking this. To specify any linear transformations between two vector spaces one needs only specify the action of the map on a basis. So, let $\displaystyle \{x_1,\cdots,x_m\}$ be a basis for $\displaystyle Z$ now since $\displaystyle x_0$ is independent of this set you know that $\displaystyle \{x_0,x_1,\cdots,x_m\}$ can be extended to some basis $\displaystyle \{x_0,x_1,\cdots,x_m,x_{m+1},\cdots,x_{n-1}\}$ for $\displaystyle X$ and define your linear functional however you want, perhaps $\displaystyle \varphi:X\to F$ given by $\displaystyle \varphi(x_k)=\delta_{k,0}$ (the Kronecker delta function) and extend by linearity.