Using 2x2 Matrices for Plane Transformations - Pre-Calculus
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What is the matrix for reflection across the $y$-axis?
What is the matrix for reflection across the $y$-axis?
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$\begin{pmatrix}-1&0\0&1\end{pmatrix}$. Negates $x$-coordinate while keeping $y$ unchanged.
$\begin{pmatrix}-1&0\0&1\end{pmatrix}$. Negates $x$-coordinate while keeping $y$ unchanged.
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Under $A=\begin{pmatrix}1&2\0&1\end{pmatrix}$, what is the image of $(3,4)$?
Under $A=\begin{pmatrix}1&2\0&1\end{pmatrix}$, what is the image of $(3,4)$?
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$(11,4)$. $\begin{pmatrix}1&2\0&1\end{pmatrix}\begin{pmatrix}3\4\end{pmatrix}=\begin{pmatrix}3+8\0+4\end{pmatrix}=\begin{pmatrix}11\4\end{pmatrix}$.
$(11,4)$. $\begin{pmatrix}1&2\0&1\end{pmatrix}\begin{pmatrix}3\4\end{pmatrix}=\begin{pmatrix}3+8\0+4\end{pmatrix}=\begin{pmatrix}11\4\end{pmatrix}$.
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For $A=\begin{pmatrix}3&0\0&-2\end{pmatrix}$, what are $\det(A)$ and $|\det(A)|$?
For $A=\begin{pmatrix}3&0\0&-2\end{pmatrix}$, what are $\det(A)$ and $|\det(A)|$?
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$\det(A)=-6$ and $|\det(A)|=6$. For diagonal matrices, $\det = $ product of diagonal entries.
$\det(A)=-6$ and $|\det(A)|=6$. For diagonal matrices, $\det = $ product of diagonal entries.
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For $A=\begin{pmatrix}0&-1\1&0\end{pmatrix}$, what is the image of $(x,y)$?
For $A=\begin{pmatrix}0&-1\1&0\end{pmatrix}$, what is the image of $(x,y)$?
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$(-y,,x)$. This matrix rotates by $90°$ counterclockwise.
$(-y,,x)$. This matrix rotates by $90°$ counterclockwise.
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What does $|\det(A)|$ represent for a $2\times 2$ matrix acting on the plane?
What does $|\det(A)|$ represent for a $2\times 2$ matrix acting on the plane?
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Area scale factor: $\text{new area}=|\det(A)|\cdot\text{old area}$. The absolute value of the determinant scales areas by that factor.
Area scale factor: $\text{new area}=|\det(A)|\cdot\text{old area}$. The absolute value of the determinant scales areas by that factor.
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What does a negative determinant ($\det(A)<0$) tell you about orientation?
What does a negative determinant ($\det(A)<0$) tell you about orientation?
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Orientation is reversed (a reflection occurs). Negative determinant flips the plane's orientation.
Orientation is reversed (a reflection occurs). Negative determinant flips the plane's orientation.
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What does $\det(A)=0$ tell you about the transformation in $\mathbb{R}^2$?
What does $\det(A)=0$ tell you about the transformation in $\mathbb{R}^2$?
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It collapses area to $0$ (not invertible). Zero determinant means the transformation is singular.
It collapses area to $0$ (not invertible). Zero determinant means the transformation is singular.
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What is the image of $(x,y)$ under $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
What is the image of $(x,y)$ under $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
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$(x',y')=(ax+by,,cx+dy)$. Matrix multiplication: first row gives $x'$, second row gives $y'$.
$(x',y')=(ax+by,,cx+dy)$. Matrix multiplication: first row gives $x'$, second row gives $y'$.
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What are the images of the basis vectors $(1,0)$ and $(0,1)$ under $A$?
What are the images of the basis vectors $(1,0)$ and $(0,1)$ under $A$?
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$(1,0)\mapsto(a,c)$ and $(0,1)\mapsto(b,d)$. Columns of $A$ are images of standard basis vectors.
$(1,0)\mapsto(a,c)$ and $(0,1)\mapsto(b,d)$. Columns of $A$ are images of standard basis vectors.
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What is the matrix for a counterclockwise rotation by angle $\theta$?
What is the 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 rotates points counterclockwise by $\theta$.
$\begin{pmatrix}\cos\theta&-\sin\theta\sin\theta&\cos\theta\end{pmatrix}$. Standard rotation matrix rotates points counterclockwise by $\theta$.
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What is $\det!\left(\begin{pmatrix}\cos\theta&-\sin\theta\sin\theta&\cos\theta\end{pmatrix}\right)$?
What is $\det!\left(\begin{pmatrix}\cos\theta&-\sin\theta\sin\theta&\cos\theta\end{pmatrix}\right)$?
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$1$. $\cos^2\theta + \sin^2\theta = 1$, so determinant equals $1$.
$1$. $\cos^2\theta + \sin^2\theta = 1$, so determinant equals $1$.
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What is the matrix for reflection across the $x$-axis?
What is the matrix for reflection across the $x$-axis?
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$\begin{pmatrix}1&0\0&-1\end{pmatrix}$. Negates $y$-coordinate while keeping $x$ unchanged.
$\begin{pmatrix}1&0\0&-1\end{pmatrix}$. Negates $y$-coordinate while keeping $x$ unchanged.
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What is the matrix for reflection across the line $y=x$?
What is the 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.
$\begin{pmatrix}0&1\1&0\end{pmatrix}$. Swaps $x$ and $y$ coordinates.
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What is the matrix for a horizontal shear with factor $k$ (so $(x,y)\mapsto(x+ky,y)$)?
What is the matrix for a horizontal shear with factor $k$ (so $(x,y)\mapsto(x+ky,y)$)?
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$\begin{pmatrix}1&k\0&1\end{pmatrix}$. Adds $k$ times $y$ to $x$, leaving $y$ unchanged.
$\begin{pmatrix}1&k\0&1\end{pmatrix}$. Adds $k$ times $y$ to $x$, leaving $y$ unchanged.
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What is the matrix for a vertical shear with factor $k$ (so $(x,y)\mapsto(x,y+kx)$)?
What is the matrix for a vertical shear with factor $k$ (so $(x,y)\mapsto(x,y+kx)$)?
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$\begin{pmatrix}1&0\k&1\end{pmatrix}$. Adds $k$ times $x$ to $y$, leaving $x$ unchanged.
$\begin{pmatrix}1&0\k&1\end{pmatrix}$. Adds $k$ times $x$ to $y$, leaving $x$ unchanged.
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What is the matrix for scaling by $s_x$ in $x$ and $s_y$ in $y$?
What is the matrix for scaling by $s_x$ in $x$ and $s_y$ in $y$?
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$\begin{pmatrix}s_x&0\0&s_y\end{pmatrix}$. Diagonal matrix scales each axis independently.
$\begin{pmatrix}s_x&0\0&s_y\end{pmatrix}$. Diagonal matrix scales each axis independently.
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Find $\det!\left(\begin{pmatrix}2&-1\3&4\end{pmatrix}\right)$.
Find $\det!\left(\begin{pmatrix}2&-1\3&4\end{pmatrix}\right)$.
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$11$. $2 \cdot 4 - (-1) \cdot 3 = 8 + 3 = 11$.
$11$. $2 \cdot 4 - (-1) \cdot 3 = 8 + 3 = 11$.
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A region has area $6$. After $A$ with $\det(A)=-3$, what is the new area?
A region has area $6$. After $A$ with $\det(A)=-3$, what is the new area?
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$18$. $|\det(A)| = |-3| = 3$, so area scales by $3$: $6 \times 3 = 18$.
$18$. $|\det(A)| = |-3| = 3$, so area scales by $3$: $6 \times 3 = 18$.
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What is the area of the parallelogram spanned by $u=(2,1)$ and $v=(5,3)$?
What is the area of the parallelogram spanned by $u=(2,1)$ and $v=(5,3)$?
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$|2\cdot 3-5\cdot 1|=1$. Area equals $|\det|$ of matrix with $u$ and $v$ as columns.
$|2\cdot 3-5\cdot 1|=1$. Area equals $|\det|$ of matrix with $u$ and $v$ as columns.
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What is the determinant of $A=\begin{pmatrix}3&1\2&4\end{pmatrix}$?
What is the determinant of $A=\begin{pmatrix}3&1\2&4\end{pmatrix}$?
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$10$. $\det = 3(4) - 1(2) = 12 - 2 = 10$.
$10$. $\det = 3(4) - 1(2) = 12 - 2 = 10$.
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What does a $2\times 2$ matrix $A$ do to a vector $(x,y)$ in the plane?
What does a $2\times 2$ matrix $A$ do to a vector $(x,y)$ in the plane?
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$A\begin{pmatrix}x\y\end{pmatrix}$ gives the transformed vector. Matrix multiplication transforms the position vector.
$A\begin{pmatrix}x\y\end{pmatrix}$ gives the transformed vector. Matrix multiplication transforms the position vector.
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What is the area scale factor of a linear transformation with matrix $A$?
What is the area scale factor of a linear transformation with matrix $A$?
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$|\det(A)|$. Absolute value of determinant measures how areas change.
$|\det(A)|$. Absolute value of determinant measures how areas change.
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What does $\det(A)<0$ indicate about a plane transformation?
What does $\det(A)<0$ indicate about a plane transformation?
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Orientation is reversed (a reflection occurs). Negative determinant flips orientation of shapes.
Orientation is reversed (a reflection occurs). Negative determinant flips orientation of shapes.
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What does $\det(A)=0$ indicate about a plane transformation?
What does $\det(A)=0$ indicate about a plane transformation?
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Area collapses to $0$ (not one-to-one; not invertible). Zero determinant means transformation squashes plane to a line.
Area collapses to $0$ (not one-to-one; not invertible). Zero determinant means transformation squashes plane to a line.
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What is the geometric meaning of $|\det(A)|$ for the unit square under $A$?
What is the geometric meaning of $|\det(A)|$ for the unit square under $A$?
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Area of the image parallelogram of the unit square. Unit square transforms to parallelogram with this area.
Area of the image parallelogram of the unit square. Unit square transforms to parallelogram with this area.
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What is the area scale factor for $A=\begin{pmatrix}-2&0\0&5\end{pmatrix}$?
What is the area scale factor for $A=\begin{pmatrix}-2&0\0&5\end{pmatrix}$?
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$10$. $|\det| = |(-2)(5) - 0(0)| = |-10| = 10$.
$10$. $|\det| = |(-2)(5) - 0(0)| = |-10| = 10$.
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What is the image of $(2,-1)$ under $A=\begin{pmatrix}1&3\0&-2\end{pmatrix}$?
What is the image of $(2,-1)$ under $A=\begin{pmatrix}1&3\0&-2\end{pmatrix}$?
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$(-1,,2)$. $A\begin{pmatrix}2\-1\end{pmatrix} = \begin{pmatrix}1(2)+3(-1)\0(2)+(-2)(-1)\end{pmatrix} = \begin{pmatrix}-1\2\end{pmatrix}$.
$(-1,,2)$. $A\begin{pmatrix}2\-1\end{pmatrix} = \begin{pmatrix}1(2)+3(-1)\0(2)+(-2)(-1)\end{pmatrix} = \begin{pmatrix}-1\2\end{pmatrix}$.
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What is the matrix for a reflection across the $x$-axis?
What is the matrix for a reflection across the $x$-axis?
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$\begin{pmatrix}1&0\0&-1\end{pmatrix}$. Keeps $x$ same, negates $y$.
$\begin{pmatrix}1&0\0&-1\end{pmatrix}$. Keeps $x$ same, negates $y$.
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What is the matrix for a reflection across the $y$-axis?
What is the matrix for a reflection across the $y$-axis?
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$\begin{pmatrix}-1&0\0&1\end{pmatrix}$. Negates $x$, keeps $y$ same.
$\begin{pmatrix}-1&0\0&1\end{pmatrix}$. Negates $x$, keeps $y$ same.
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What is the matrix for a $90^\circ$ counterclockwise rotation about the origin?
What is the matrix for a $90^\circ$ counterclockwise rotation about the origin?
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$\begin{pmatrix}0&-1\1&0\end{pmatrix}$. Maps $(1,0)$ to $(0,1)$ and $(0,1)$ to $(-1,0)$.
$\begin{pmatrix}0&-1\1&0\end{pmatrix}$. Maps $(1,0)$ to $(0,1)$ and $(0,1)$ to $(-1,0)$.
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