Science:Math Exam Resources/Courses/MATH307/December 2005/Question 05 (b)

MATH307 December 2005

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Question 05 (b)

Let X be the subspace in three dimensional space $\mathbb {R} ^{3}$ containing all vectors perpendicular to ${\begin{bmatrix}1\\1\\1\end{bmatrix}}$ Let $L$ be the linear transformation defined by the first projective vector onto the $x-y$ plane and then projecting the resulting vector onto X. Find the $3x3$ matrix that represents $L$ as a linear transformation from $\mathbb {R} ^{3}$ to $\mathbb {R} ^{3}$

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Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We need to find the projection matrix. (Note: [ ] denote matrices) [projecting vector that projects resulting vector onto X] [projecting vector that projects onto x-y]${\bar {x}}$ To find the projecting vector that projects onto x-y basis vectors are ${\begin{bmatrix}1\\0\end{bmatrix}},{\begin{bmatrix}0\\1\end{bmatrix}}$ $P=A^{T}(A^{T}A)^{-1}A^{T}$ where A = 2x2 identity matrix (from our two vectors above) Therefore: P = 2x2 identity matrix To find the projecting vector that projects resulting vector onto X: projecting vector onto the span(a) where $a={\begin{bmatrix}1\\1\\1\end{bmatrix}}$ $A={\frac {aa^{T}}{\left\|a\right\|^{2}}}={\frac {1}{3}}{\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}}$ The projecting vector to the orthogonal space = $I_{3}-A$ $={\begin{bmatrix}0&-1&-1\\-1&0&-1\\-1&-1&0\end{bmatrix}}$ P*A = $={\begin{bmatrix}0&-1&-1\\-1&0&-1\\-1&-1&0\end{bmatrix}}$ which is our required projecting matrix</p> <p></p>