## Linear Transformation to 1-Dimensional Vector Space and Its Kernel

## Problem 329

Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation.

Prove the followings.

**(a)** The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$.

**(b)** Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the nullspace $\calN(T)$ of $T$.

Let $\mathbf{w}$ be the $n$-dimensional vector that is not in $\calN(T)$. Then

\[B’=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}, \mathbf{w}\}\]
is a basis of $\R^n$.

**(c)** Each vector $\mathbf{u}\in \R^n$ can be expressed as

\[\mathbf{u}=\mathbf{v}+\frac{T(\mathbf{u})}{T(\mathbf{w})}\mathbf{w}\]
for some vector $\mathbf{v}\in \calN(T)$.