1. ## Null Space

Please can you guys help me to solve the following questions

Q1.if Z is an (n-1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null space of suitable linear functional f on X, which is uniquely determined to within a scalar multiple.

2. Originally Posted by kinkong
Please can you guys help me to solve the following questions

Q1.if Z is an (n-1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null space of suitable linear functional f on X, which is uniquely determined to within a scalar multiple.
Use the exact same logic as my other post. Take a basis $\{x_1,\cdots,x_{n-1}\}$ for $Z$ and extend it to a basis $\{x_1,\cdots,x_n\}$ for $X$ and define $\varphi:X\to F$ by $\varphi(x_k)=\delta_{k,n}$. To prove it's unique up to a scaling factor note that any linear functional $\psi$ must have the same form except $\psi(x_n)\ne 0$ may be different and so evidently $\varphi=\frac{1}{\psi(x_n)}\psi$

3. note that the linear functional φ in this case is the dual vector to the basis element xn.

note also that since n = dim(X) = null(φ) + rank(φ) = null(φ) + 1, dim(null(φ)) = n-1.

it is a theorem that every hyper-space (or hyper-plane) arises in this fashion.