Previous in Matrices and Determinants Next in Matrices and Determinants
Question Number 124170 by Pengu last updated on 01/Dec/20
![I am learning linear algebra and have a qustion in regards to nullspace. So, Ax=0. If A= [(1,1),(0,0) ], this is simply solved for x= [(x_1 ),(x_2 ) ] by: [(1,1,0),(0,0,0) ]→x= [((−1)),(1) ]c, ∀c∈R Since this is for all x∈R^2 , the null space falls on a line (in this case). My question is, in higher dimensions, can the null space be any dimension up to the dimension n? So, in R^4 , can the null space be a point, line, plane, or volume? Furthermore, can you have multiple variations of planes, points, etc. that make up the null space? Thanks!](Q124170.png)
$$ \\ $$$$\mathrm{I}\:\mathrm{am}\:\mathrm{learning}\:\mathrm{linear}\:\mathrm{algebra}\:\mathrm{and}\:\mathrm{have}\:\mathrm{a} \\ $$$$\mathrm{qustion}\:\mathrm{in}\:\mathrm{regards}\:\mathrm{to}\:\mathrm{nullspace}. \\ $$$$\: \\ $$$$\mathrm{So},\:\boldsymbol{{Ax}}=\mathrm{0}. \\ $$$$\mathrm{If}\:\boldsymbol{{A}}=\begin{bmatrix}{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{0}}\end{bmatrix},\:\mathrm{this}\:\mathrm{is}\:\mathrm{simply}\:\mathrm{solved}\:\mathrm{for} \\ $$$$\boldsymbol{{x}}=\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\end{bmatrix} \\ $$$$\mathrm{by}: \\ $$$$\begin{bmatrix}{\mathrm{1}}&{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\end{bmatrix}\rightarrow\boldsymbol{{x}}=\begin{bmatrix}{−\mathrm{1}}\\{\mathrm{1}}\end{bmatrix}{c},\:\forall{c}\in\mathbb{R} \\ $$$$\mathrm{Since}\:\mathrm{this}\:\mathrm{is}\:\mathrm{for}\:\mathrm{all}\:\boldsymbol{{x}}\in\mathbb{R}^{\mathrm{2}} ,\:\mathrm{the}\:\mathrm{null}\:\mathrm{space} \\ $$$$\mathrm{falls}\:\mathrm{on}\:\mathrm{a}\:\mathrm{line}\:\left(\mathrm{in}\:\mathrm{this}\:\mathrm{case}\right). \\ $$$$\: \\ $$$$\mathrm{My}\:\mathrm{question}\:\mathrm{is},\:\mathrm{in}\:\mathrm{higher}\:\mathrm{dimensions}, \\ $$$$\mathrm{can}\:\mathrm{the}\:\mathrm{null}\:\mathrm{space}\:\mathrm{be}\:\mathrm{any}\:\mathrm{dimension}\:\mathrm{up} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{dimension}\:{n}?\:\mathrm{So},\:\mathrm{in}\:\mathbb{R}^{\mathrm{4}} ,\:\mathrm{can}\:\mathrm{the} \\ $$$$\mathrm{null}\:\mathrm{space}\:\mathrm{be}\:\mathrm{a}\:\mathrm{point},\:\mathrm{line},\:\mathrm{plane},\:\mathrm{or}\:\mathrm{volume}? \\ $$$$\mathrm{Furthermore},\:\mathrm{can}\:\mathrm{you}\:\mathrm{have}\:{multiple}\:\mathrm{variations} \\ $$$$\mathrm{of}\:\mathrm{planes},\:\mathrm{points},\:\mathrm{etc}.\:\mathrm{that}\:\mathrm{make}\:\mathrm{up}\:\mathrm{the}\:\mathrm{null}\:\mathrm{space}? \\ $$$$\: \\ $$$$\mathrm{Thanks}! \\ $$