Using 2x2 Matrices for Plane Transformations - Geometry
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What is the image of $(4,7)$ under reflection across $y=x$?
What is the image of $(4,7)$ under reflection across $y=x$?
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$(7,4)$. Reflection across $y=x$ swaps coordinates.
$(7,4)$. Reflection across $y=x$ swaps coordinates.
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What is the area of the parallelogram spanned by vectors $\vec{u}$ and $\vec{v}$ in $\mathbb{R}^2$?
What is the area of the parallelogram spanned by vectors $\vec{u}$ and $\vec{v}$ in $\mathbb{R}^2$?
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$\left|\det!\left(\begin{pmatrix}u_1&v_1\u_2&v_2\end{pmatrix}\right)\right|$. Parallelogram area equals absolute value of cross product.
$\left|\det!\left(\begin{pmatrix}u_1&v_1\u_2&v_2\end{pmatrix}\right)\right|$. Parallelogram area equals absolute value of cross product.
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What is $\det!\left(\begin{pmatrix}k&0\0&1\end{pmatrix}\right)$ for a horizontal stretch by $k$?
What is $\det!\left(\begin{pmatrix}k&0\0&1\end{pmatrix}\right)$ for a horizontal stretch by $k$?
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$k$. Horizontal stretch scales area by factor $k$.
$k$. Horizontal stretch scales area by factor $k$.
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What is the image of $(-5,2)$ under reflection across the $x$-axis?
What is the image of $(-5,2)$ under reflection across the $x$-axis?
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$(-5,-2)$. Reflection across $x$-axis negates $y$-coordinate.
$(-5,-2)$. Reflection across $x$-axis negates $y$-coordinate.
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What is the image of $(-5,2)$ under reflection across the $y$-axis?
What is the image of $(-5,2)$ under reflection across the $y$-axis?
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$(5,2)$. Reflection across $y$-axis negates $x$-coordinate.
$(5,2)$. Reflection across $y$-axis negates $x$-coordinate.
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What is the area scale factor for $A=\begin{pmatrix}0&-2\1&0\end{pmatrix}$?
What is the area scale factor for $A=\begin{pmatrix}0&-2\1&0\end{pmatrix}$?
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$|\det(A)|=2$. $|0 \times 0 - (-2) \times 1| = 2$
$|\det(A)|=2$. $|0 \times 0 - (-2) \times 1| = 2$
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What is the image of $(1,0)$ under $A=\begin{pmatrix}2&3\1&4\end{pmatrix}$?
What is the image of $(1,0)$ under $A=\begin{pmatrix}2&3\1&4\end{pmatrix}$?
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$(2,1)$. First column gives image of $(1,0)$.
$(2,1)$. First column gives image of $(1,0)$.
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What is the image of $(0,1)$ under $A=\begin{pmatrix}2&3\1&4\end{pmatrix}$?
What is the image of $(0,1)$ under $A=\begin{pmatrix}2&3\1&4\end{pmatrix}$?
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$(3,4)$. Second column gives image of $(0,1)$.
$(3,4)$. Second column gives image of $(0,1)$.
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If $\det(A)=-2$ and $\det(B)=-3$, what is $\det(AB)$?
If $\det(A)=-2$ and $\det(B)=-3$, what is $\det(AB)$?
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$6$. $(-2) \times (-3) = 6$
$6$. $(-2) \times (-3) = 6$
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What does $|\det(A)|$ represent for a $2\times 2$ linear transformation of the plane?
What does $|\det(A)|$ represent for a $2\times 2$ linear transformation of the plane?
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Area scale factor (multiplier) for all regions. The absolute value of determinant scales all areas uniformly.
Area scale factor (multiplier) for all regions. The absolute value of determinant scales all areas uniformly.
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What does the sign of $\det(A)$ indicate for a plane transformation?
What does the sign of $\det(A)$ indicate for a plane transformation?
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$\det(A)>0$ preserves orientation; $\det(A)<0$ reverses it. Positive preserves clockwise/counterclockwise; negative flips it.
$\det(A)>0$ preserves orientation; $\det(A)<0$ reverses it. Positive preserves clockwise/counterclockwise; negative flips it.
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What is the determinant of $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
What is the determinant of $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
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$\det(A)=ad-bc$. Formula: first diagonal product minus second diagonal product.
$\det(A)=ad-bc$. Formula: first diagonal product minus second diagonal product.
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What is the image of a vector $\begin{pmatrix}x\y\end{pmatrix}$ under $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
What is the image of a vector $\begin{pmatrix}x\y\end{pmatrix}$ under $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
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$A\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}ax+by\cx+dy\end{pmatrix}$. Matrix multiplication applies the transformation to the vector.
$A\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}ax+by\cx+dy\end{pmatrix}$. Matrix multiplication applies the transformation to the vector.
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What is the determinant of a uniform dilation matrix $kI$ in $\mathbb{R}^2$?
What is the determinant of a uniform dilation matrix $kI$ in $\mathbb{R}^2$?
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$\det(kI)=k^2$. Uniform scaling in 2D scales area by $k^2$.
$\det(kI)=k^2$. Uniform scaling in 2D scales area by $k^2$.
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What does $\det(A)=0$ imply about the image of the plane under $A$?
What does $\det(A)=0$ imply about the image of the plane under $A$?
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It collapses to a line or point (not invertible). Zero determinant means the transformation is not invertible.
It collapses to a line or point (not invertible). Zero determinant means the transformation is not invertible.
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What is the area of the image of a region with area $K$ under matrix $A$?
What is the area of the image of a region with area $K$ under matrix $A$?
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$K' = |\det(A)|,K$. Area scales by the absolute value of the determinant.
$K' = |\det(A)|,K$. Area scales by the absolute value of the determinant.
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What is the determinant of the identity matrix $I=\begin{pmatrix}1&0\0&1\end{pmatrix}$?
What is the determinant of the identity matrix $I=\begin{pmatrix}1&0\0&1\end{pmatrix}$?
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$\det(I)=1$. Identity transformation preserves all areas, so determinant is 1.
$\det(I)=1$. Identity transformation preserves all areas, so determinant is 1.
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What is the $2\times 2$ matrix for reflection across the $x$-axis?
What is the $2\times 2$ matrix for reflection across the $x$-axis?
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$\begin{pmatrix}1&0\0&-1\end{pmatrix}$. Flips $y$-coordinate while keeping $x$-coordinate unchanged.
$\begin{pmatrix}1&0\0&-1\end{pmatrix}$. Flips $y$-coordinate while keeping $x$-coordinate unchanged.
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What is the $2\times 2$ matrix for reflection across the $y$-axis?
What is the $2\times 2$ matrix for reflection across the $y$-axis?
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$\begin{pmatrix}-1&0\0&1\end{pmatrix}$. Flips $x$-coordinate while keeping $y$-coordinate unchanged.
$\begin{pmatrix}-1&0\0&1\end{pmatrix}$. Flips $x$-coordinate while keeping $y$-coordinate unchanged.
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What is the $2\times 2$ matrix for reflection across the line $y=x$?
What is the $2\times 2$ matrix for reflection across the line $y=x$?
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$\begin{pmatrix}0&1\1&0\end{pmatrix}$. Swaps $x$ and $y$ coordinates to reflect across diagonal.
$\begin{pmatrix}0&1\1&0\end{pmatrix}$. Swaps $x$ and $y$ coordinates to reflect across diagonal.
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What is the $2\times 2$ matrix for a $90^\circ$ counterclockwise rotation?
What is the $2\times 2$ matrix for a $90^\circ$ counterclockwise rotation?
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$\begin{pmatrix}0&-1\1&0\end{pmatrix}$. Rotates point $(x,y)$ to $(-y,x)$ counterclockwise.
$\begin{pmatrix}0&-1\1&0\end{pmatrix}$. Rotates point $(x,y)$ to $(-y,x)$ counterclockwise.
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What is the $2\times 2$ matrix for a $180^\circ$ rotation about the origin?
What is the $2\times 2$ matrix for a $180^\circ$ rotation about the origin?
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$\begin{pmatrix}-1&0\0&-1\end{pmatrix}$. Rotates point $(x,y)$ to $(-x,-y)$ around origin.
$\begin{pmatrix}-1&0\0&-1\end{pmatrix}$. Rotates point $(x,y)$ to $(-x,-y)$ around origin.
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What is the $2\times 2$ matrix for a $270^\circ$ counterclockwise rotation?
What is the $2\times 2$ matrix for a $270^\circ$ counterclockwise rotation?
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$\begin{pmatrix}0&1\-1&0\end{pmatrix}$. Rotates point $(x,y)$ to $(y,-x)$ counterclockwise.
$\begin{pmatrix}0&1\-1&0\end{pmatrix}$. Rotates point $(x,y)$ to $(y,-x)$ counterclockwise.
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What is the $2\times 2$ matrix for a counterclockwise rotation by angle $\theta$?
What is the $2\times 2$ matrix for a counterclockwise rotation by angle $\theta$?
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$\begin{pmatrix}\cos\theta&-\sin\theta\sin\theta&\cos\theta\end{pmatrix}$. Standard rotation matrix using trigonometric identities.
$\begin{pmatrix}\cos\theta&-\sin\theta\sin\theta&\cos\theta\end{pmatrix}$. Standard rotation matrix using trigonometric identities.
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What is the determinant of a rotation matrix $R_\theta$?
What is the determinant of a rotation matrix $R_\theta$?
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$\det(R_\theta)=1$. Rotations preserve areas, so determinant equals 1.
$\det(R_\theta)=1$. Rotations preserve areas, so determinant equals 1.
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What is the determinant of any reflection matrix in the plane?
What is the determinant of any reflection matrix in the plane?
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$\det=-1$. Reflections preserve area but reverse orientation.
$\det=-1$. Reflections preserve area but reverse orientation.
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What is the matrix for a horizontal stretch by factor $k$ (scale $x$ only)?
What is the matrix for a horizontal stretch by factor $k$ (scale $x$ only)?
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$\begin{pmatrix}k&0\0&1\end{pmatrix}$. Stretches $x$-coordinate by factor $k$, leaves $y$ unchanged.
$\begin{pmatrix}k&0\0&1\end{pmatrix}$. Stretches $x$-coordinate by factor $k$, leaves $y$ unchanged.
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What is the matrix for a vertical stretch by factor $k$ (scale $y$ only)?
What is the matrix for a vertical stretch by factor $k$ (scale $y$ only)?
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$\begin{pmatrix}1&0\0&k\end{pmatrix}$. Stretches $y$-coordinate by factor $k$, leaves $x$ unchanged.
$\begin{pmatrix}1&0\0&k\end{pmatrix}$. Stretches $y$-coordinate by factor $k$, leaves $x$ unchanged.
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What is the matrix for a uniform dilation by factor $k$ about the origin?
What is the matrix for a uniform dilation by factor $k$ about the origin?
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$\begin{pmatrix}k&0\0&k\end{pmatrix}$. Scales both coordinates by the same factor $k$.
$\begin{pmatrix}k&0\0&k\end{pmatrix}$. Scales both coordinates by the same factor $k$.
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What is the matrix for a shear parallel to the $x$-axis: $(x,y)\mapsto(x+ky,y)$?
What is the matrix for a shear parallel to the $x$-axis: $(x,y)\mapsto(x+ky,y)$?
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$\begin{pmatrix}1&k\0&1\end{pmatrix}$. Shears horizontally: $x$ gets shifted by $k$ times $y$.
$\begin{pmatrix}1&k\0&1\end{pmatrix}$. Shears horizontally: $x$ gets shifted by $k$ times $y$.
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