每週問題 June 20, 2016

計算一個線性變換的秩。

Let \{\mathbf{q}_1,\mathbf{q}_2,\mathbf{q}_3\} be an orthonormal set in \mathbb{R}^3 and \mathbf{q}_3 be the cross product of \mathbf{q}_1 and \mathbf{q}_2, i.e., \mathbf{q}_3=\mathbf{q}_1\times\mathbf{q}_2. A linear transformation T:\mathbb{R}^3\to\mathbb{R}^3 is defined by

T(\mathbf{x})=\mathbf{x}\times \mathbf{q}_1+(\mathbf{q}_2^T\mathbf{x})\mathbf{q}_1.

Determine the rank of T.

 
參考解答:

計算 \mathbb{R}^3 的單範正交基底 (orthonormal basis) \{\mathbf{q}_1,\mathbf{q}_2,\mathbf{q}_3\} 的像 (image):

\begin{aligned} T(\mathbf{q}_1)&=\mathbf{q}_1\times \mathbf{q}_1+(\mathbf{q}_2^T\mathbf{q}_1)\mathbf{q}_1=\mathbf{0}\\ T(\mathbf{q}_2)&=\mathbf{q}_2\times \mathbf{q}_1+(\mathbf{q}_2^T\mathbf{q}_2)\mathbf{q}_1=-\mathbf{q}_3+\mathbf{q}_1\\ T(\mathbf{q}_3)&=\mathbf{q}_3\times \mathbf{q}_1+(\mathbf{q}_2^T\mathbf{q}_3)\mathbf{q}_1=\mathbf{q}_2. \end{aligned}

因此,線性變換 T 的值域 \hbox{ran}(T) 的一組基底為 \{\mathbf{q}_1-\mathbf{q}_3,\mathbf{q}_2\},可知 \hbox{rank}T=\dim \hbox{ran}(T)=2

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