# Thread: Vector space and null space

1. ## Vector space and null space

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

2. Originally Posted by kinkong
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 $\{x_1,\cdots,x_m\}$ be a basis for $Z$ now since $x_0$ is independent of this set you know that $\{x_0,x_1,\cdots,x_m\}$ can be extended to some basis $\{x_0,x_1,\cdots,x_m,x_{m+1},\cdots,x_{n-1}\}$ for $X$ and define your linear functional however you want, perhaps $\varphi:X\to F$ given by $\varphi(x_k)=\delta_{k,0}$ (the Kronecker delta function) and extend by linearity.