27. Consider the matrices A through E below. \begin{align*} A &= \begin{bmatrix} 0.6 & 0.8\\ 0.8 & -0.6 \end{bmatrix}, & B &= \begin{bmatrix} 3 & 0\\ 0 & 3 \end{bmatrix},\\ C &= \begin{bmatrix} 0.36 & -0.48\\ -0.48 & 0.64 \end{bmatrix}, & D &= \begin{bmatrix} -0.8 & 0.6\\ -0.6 & -0.8 \end{bmatrix},\\ E &= \begin{bmatrix} 1 & 0\\ -1 & 1 \end{bmatrix} \end{align*} Fill in the blanks in the sentences below. We are told that there is a solution in each case. Matrix ______ represents a scaling. Matrix ______ represents an orthogonal projection. Matrix ______ represents a shear. Matrix ______ represents a reflection. Matrix ______ represents a rotation.
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Matrix B: This matrix represents an orthogonal projection because it projects the coordinates onto a lower-dimensional subspace. Matrix C: This matrix represents a shear because it shifts the coordinates along one axis while keeping the other axis fixed. Matrix Show more…
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Consider the matrices $A$ through $E$ below. \[\begin{aligned}A=\left[\begin{array}{rr}0.6 & 0.8 \\0.8 & -0.6\end{array}\right], \quad B=\left[\begin{array}{ll}3 & 0 \\0 & 3\end{array}\right] \\C=\left[\begin{array}{rr}0.36 & -0.48 \\ -0.48 & 0.64\end{array}\right], \quad D=\left[\begin{array}{rr}-0.8 & 0.6 \\-0.6 & -0.8\end{array}\right] \\ E=\left[\begin{array}{rr}1 & 0 \\-1 & 1\end{array}\right]\end{aligned}\] Fill in the blanks in the sentences below. We are told that there is a solution in each case. Matrix ___ represents a scaling. Matrix ___ represents an orthogonal projection. Matrix ___ represents a shear. Matrix ___ represents a reflection. Matrix ___ represents a rotation.
Linear Transformations
Linear Transformations in Geometry
Each of the linear transformations in parts (a) through (e) corresponds to one (and only one) of the matrices $A$ through $J$. Match them up. a. Scaling b. Shear $\mathbf{c}_{*} \quad$ Rotation d. Orthogonal projection e. $\quad$ Reflection $A=\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right], \quad B=\left[\begin{array}{ll}2 & 1 \\ 1 & 0\end{array}\right], \quad C=\left[\begin{array}{rr}-0.6 & 0.8 \\ -0.8 & -0.6\end{array}\right]$ $D=\left[\begin{array}{ll}7 & 0 \\ 0 & 7\end{array}\right], \quad E=\left[\begin{array}{rr}1 & 0 \\ -3 & 1\end{array}\right], \quad F=\left[\begin{array}{rr}0.6 & 0.8 \\ 0.8 & -0.6\end{array}\right]$ $\begin{aligned} G=\left[\begin{array}{ll}0.6 & 0.6 \\ 0.8 & 0.8\end{array}\right], \quad H=\left[\begin{array}{rr}2 & -1 \\ 1 & 2\end{array}\right], \quad I=\left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right] \\ J &=\left[\begin{array}{ll}0.8 & -0.6 \\ 0.6 & -0.8\end{array}\right] \end{aligned}$
7. Orthogonally diagonalize the matrices by finding an orthogonal matrix Q and a diagonal matrix D such that QT AQ = D. A = [ 1 0 -1; 0 1 0; -1 0 1] B = [ 2 0 0 1; 0 1 0 0; 0 0 1 0; 1 0 0 2]
Adi S.
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