Algebra and Geometry | Part IA, 2001

Let lxl_{\mathbf{x}} denote the straight line through x\mathbf{x} with directional vector u0\mathbf{u} \neq \mathbf{0}

lx={yR3:y=x+λu,λR}l_{\mathbf{x}}=\left\{\mathbf{y} \in \mathbb{R}^{3}: \mathbf{y}=\mathbf{x}+\lambda \mathbf{u}, \lambda \in \mathbb{R}\right\}

Show that l0l_{\mathbf{0}} is a subspace of R3\mathbb{R}^{3} and show that lx1=lx2x1=x2+λul_{\mathbf{x}_{1}}=l_{\mathbf{x}_{2}} \Leftrightarrow \mathbf{x}_{\mathbf{1}}=\mathbf{x}_{2}+\lambda \mathbf{u} for some λR\lambda \in \mathbb{R}.

For fixed u0\mathbf{u} \neq \mathbf{0} let L\mathcal{L} be the set of all the parallel straight lines lx(xR3)l_{\mathbf{x}}\left(\mathbf{x} \in \mathbb{R}^{3}\right) with directional vector u\mathbf{u}. On L\mathcal{L} an addition and a scalar multiplication are defined by

lx+ly=lx+y,αlx=lαx,x,yR3,αRl_{\mathbf{x}}+l_{\mathbf{y}}=l_{\mathbf{x}+\mathbf{y}}, \alpha l_{\mathbf{x}}=l_{\alpha \mathbf{x}}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^{3}, \alpha \in \mathbb{R}

Explain why these operations are well-defined. Show that the addition is associative and that there exists a zero vector which should be identified.

You may now assume that L\mathcal{L} is a vector space. If {u,b1,b2}\left\{\mathbf{u}, \mathbf{b}_{1}, \mathbf{b}_{2}\right\} is a basis for R3\mathbb{R}^{3} show that {lb1,lb2}\left\{l_{\mathbf{b}_{1}}, l_{\mathbf{b}_{2}}\right\} is a basis for L\mathcal{L}.

For u=(1,3,1)T\mathbf{u}=(1,3,-1)^{T} a linear map Φ:LL\Phi: \mathcal{L} \rightarrow \mathcal{L} is defined by

Φ(l(1,1,0)T)=l(2,4,1)T,Φ(l(1,1,0)T)=l(4,2,1)T\Phi\left(l_{(1,-1,0)^{T}}\right)=l_{(2,4,-1)^{T}}, \Phi\left(l_{(1,1,0)^{T}}\right)=l_{(-4,-2,1)^{T}}

Find the matrix AA of Φ\Phi with respect to the basis {l(1,0,0)T,l(0,1,0)T}\left\{l_{(1,0,0)^{T}}, l_{(0,1,0)^{T}}\right\}.

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