Matrix Transformations of Vectors - Pre-Calculus
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What is the dimension of the product $A\vec{x}$ if $A$ is $m\times n$ and $\vec{x}$ is $n\times 1$?
What is the dimension of the product $A\vec{x}$ if $A$ is $m\times n$ and $\vec{x}$ is $n\times 1$?
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$A\vec{x}$ is $m\times 1$. Result has rows of $A$ and columns of $\vec{x}$.
$A\vec{x}$ is $m\times 1$. Result has rows of $A$ and columns of $\vec{x}$.
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What is the entry formula for $(A\vec{x})_i$ when $A$ is $m\times n$ and $\vec{x}$ is $n\times 1$?
What is the entry formula for $(A\vec{x})_i$ when $A$ is $m\times n$ and $\vec{x}$ is $n\times 1$?
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$(A\vec{x})i=\sum{j=1}^{n}a_{ij}x_j$. Row $i$ of $A$ dot product with vector $\vec{x}$.
$(A\vec{x})i=\sum{j=1}^{n}a_{ij}x_j$. Row $i$ of $A$ dot product with vector $\vec{x}$.
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Identify the correct interpretation of $A\vec{x}$ in $\mathbb{R}^n$ when $A$ is $n\times n$.
Identify the correct interpretation of $A\vec{x}$ in $\mathbb{R}^n$ when $A$ is $n\times n$.
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$A\vec{x}$ is the image of $\vec{x}$ under the linear transformation $A$. Matrix multiplication represents a linear transformation.
$A\vec{x}$ is the image of $\vec{x}$ under the linear transformation $A$. Matrix multiplication represents a linear transformation.
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What is the $2\times 2$ matrix for scaling by factor $k$ in both directions in $\mathbb{R}^2$?
What is the $2\times 2$ matrix for scaling by factor $k$ in both directions in $\mathbb{R}^2$?
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$\begin{pmatrix}k&0\0&k\end{pmatrix}$. Diagonal matrix with $k$ scales both coordinates by $k$.
$\begin{pmatrix}k&0\0&k\end{pmatrix}$. Diagonal matrix with $k$ scales both coordinates by $k$.
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What is the $2\times 2$ matrix that reflects points across the $x$-axis in $\mathbb{R}^2$?
What is the $2\times 2$ matrix that reflects points across the $x$-axis in $\mathbb{R}^2$?
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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 $2\times 2$ matrix that reflects points across the $y$-axis in $\mathbb{R}^2$?
What is the $2\times 2$ matrix that reflects points across the $y$-axis in $\mathbb{R}^2$?
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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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What is the $2\times 2$ matrix that swaps coordinates, reflecting across the line $y=x$?
What is the $2\times 2$ matrix that swaps coordinates, reflecting 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 $2\times 2$ matrix for a counterclockwise rotation by angle $\theta$ in $\mathbb{R}^2$?
What is the $2\times 2$ matrix for a counterclockwise rotation by angle $\theta$ in $\mathbb{R}^2$?
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$\begin{pmatrix}\cos\theta&-\sin\theta\sin\theta&\cos\theta\end{pmatrix}$. Standard rotation matrix using trig functions.
$\begin{pmatrix}\cos\theta&-\sin\theta\sin\theta&\cos\theta\end{pmatrix}$. Standard rotation matrix using trig functions.
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What is the $2\times 2$ matrix for a horizontal shear with factor $k$ (sending $(x,y)$ to $(x+ky,y)$)?
What is the $2\times 2$ matrix for a horizontal shear with factor $k$ (sending $(x,y)$ to $(x+ky,y)$)?
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$\begin{pmatrix}1&k\0&1\end{pmatrix}$. Adds $k$ times $y$ to $x$, keeping $y$ unchanged.
$\begin{pmatrix}1&k\0&1\end{pmatrix}$. Adds $k$ times $y$ to $x$, keeping $y$ unchanged.
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Compute $A\vec{x}$ for $A=\begin{pmatrix}2&-1\0&3\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}4\-2\end{pmatrix}$.
Compute $A\vec{x}$ for $A=\begin{pmatrix}2&-1\0&3\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}4\-2\end{pmatrix}$.
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$\begin{pmatrix}10\-6\end{pmatrix}$. $2(4)+(-1)(-2)=10$; $0(4)+3(-2)=-6$.
$\begin{pmatrix}10\-6\end{pmatrix}$. $2(4)+(-1)(-2)=10$; $0(4)+3(-2)=-6$.
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Compute $A\vec{x}$ for $A=\begin{pmatrix}1&2\3&4\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}-1\5\end{pmatrix}$.
Compute $A\vec{x}$ for $A=\begin{pmatrix}1&2\3&4\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}-1\5\end{pmatrix}$.
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$\begin{pmatrix}9\17\end{pmatrix}$. $1(-1)+2(5)=9$; $3(-1)+4(5)=17$.
$\begin{pmatrix}9\17\end{pmatrix}$. $1(-1)+2(5)=9$; $3(-1)+4(5)=17$.
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Compute $A\vec{x}$ for $A=\begin{pmatrix}0&-1\1&0\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}2\3\end{pmatrix}$.
Compute $A\vec{x}$ for $A=\begin{pmatrix}0&-1\1&0\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}2\3\end{pmatrix}$.
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$\begin{pmatrix}-3\2\end{pmatrix}$. $0(2)+(-1)(3)=-3$; $1(2)+0(3)=2$.
$\begin{pmatrix}-3\2\end{pmatrix}$. $0(2)+(-1)(3)=-3$; $1(2)+0(3)=2$.
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Compute $A\vec{x}$ for $A=\begin{pmatrix}1&0\0&-1\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}-6\4\end{pmatrix}$.
Compute $A\vec{x}$ for $A=\begin{pmatrix}1&0\0&-1\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}-6\4\end{pmatrix}$.
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$\begin{pmatrix}-6\-4\end{pmatrix}$. $1(-6)+0(4)=-6$; $0(-6)+(-1)(4)=-4$.
$\begin{pmatrix}-6\-4\end{pmatrix}$. $1(-6)+0(4)=-6$; $0(-6)+(-1)(4)=-4$.
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Compute $A\vec{x}$ for $A=\begin{pmatrix}3&0\0&\frac{1}{2}\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}2\8\end{pmatrix}$.
Compute $A\vec{x}$ for $A=\begin{pmatrix}3&0\0&\frac{1}{2}\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}2\8\end{pmatrix}$.
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$\begin{pmatrix}6\4\end{pmatrix}$. $3(2)+0(8)=6$; $0(2)+\frac{1}{2}(8)=4$.
$\begin{pmatrix}6\4\end{pmatrix}$. $3(2)+0(8)=6$; $0(2)+\frac{1}{2}(8)=4$.
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Compute $A\vec{x}$ for $A=\begin{pmatrix}1&4\0&1\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}1\-2\end{pmatrix}$.
Compute $A\vec{x}$ for $A=\begin{pmatrix}1&4\0&1\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}1\-2\end{pmatrix}$.
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$\begin{pmatrix}-7\-2\end{pmatrix}$. $1(1)+4(-2)=-7$; $0(1)+1(-2)=-2$.
$\begin{pmatrix}-7\-2\end{pmatrix}$. $1(1)+4(-2)=-7$; $0(1)+1(-2)=-2$.
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Identify whether $A\vec{x}$ is defined for $A$ of size $3\times 2$ and $\vec{x}$ of size $2\times 1$.
Identify whether $A\vec{x}$ is defined for $A$ of size $3\times 2$ and $\vec{x}$ of size $2\times 1$.
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Defined; the product has size $3\times 1$. Columns of $A$ (2) equals rows of $\vec{x}$ (2).
Defined; the product has size $3\times 1$. Columns of $A$ (2) equals rows of $\vec{x}$ (2).
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What is the linear-combination form of $A\begin{pmatrix}x\y\end{pmatrix}$ using columns $\vec{a}_1,\vec{a}_2$ of $A$?
What is the linear-combination form of $A\begin{pmatrix}x\y\end{pmatrix}$ using columns $\vec{a}_1,\vec{a}_2$ of $A$?
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$A\begin{pmatrix}x\y\end{pmatrix}=x\vec{a}_1+y\vec{a}_2$. Matrix multiplication is a linear combination of columns.
$A\begin{pmatrix}x\y\end{pmatrix}=x\vec{a}_1+y\vec{a}_2$. Matrix multiplication is a linear combination of columns.
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What dimension must a matrix $A$ have to multiply a vector $\vec{v}$ of size $n\times 1$ on the left?
What dimension must a matrix $A$ have to multiply a vector $\vec{v}$ of size $n\times 1$ on the left?
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$A$ must be $m\times n$ so $A\vec{v}$ is $m\times 1$. Matrix cols must match vector rows for multiplication.
$A$ must be $m\times n$ so $A\vec{v}$ is $m\times 1$. Matrix cols must match vector rows for multiplication.
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What is the output type of multiplying a matrix $A$ by a column vector $\vec{v}$ (with compatible dimensions)?
What is the output type of multiplying a matrix $A$ by a column vector $\vec{v}$ (with compatible dimensions)?
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A column vector (a matrix of size $m\times 1$). Matrix-vector multiplication preserves the column structure.
A column vector (a matrix of size $m\times 1$). Matrix-vector multiplication preserves the column structure.
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What is the formula for $A\vec{v}$ when $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$ and $\vec{v}=\begin{pmatrix}x\y\end{pmatrix}$?
What is the formula for $A\vec{v}$ when $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$ and $\vec{v}=\begin{pmatrix}x\y\end{pmatrix}$?
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$A\vec{v}=\begin{pmatrix}ax+by\cx+dy\end{pmatrix}$. Each component is the dot product of a row with the vector.
$A\vec{v}=\begin{pmatrix}ax+by\cx+dy\end{pmatrix}$. Each component is the dot product of a row with the vector.
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Identify the dimension error: Can $\begin{pmatrix}1&2&3\4&5&6\end{pmatrix}\begin{pmatrix}7\8\end{pmatrix}$ be computed?
Identify the dimension error: Can $\begin{pmatrix}1&2&3\4&5&6\end{pmatrix}\begin{pmatrix}7\8\end{pmatrix}$ be computed?
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No; inner dimensions $3$ and $2$ do not match. Matrix has 3 columns but vector has 2 rows.
No; inner dimensions $3$ and $2$ do not match. Matrix has 3 columns but vector has 2 rows.
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What is $\begin{pmatrix}2&0\0&3\end{pmatrix}\begin{pmatrix}4\-1\end{pmatrix}$?
What is $\begin{pmatrix}2&0\0&3\end{pmatrix}\begin{pmatrix}4\-1\end{pmatrix}$?
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$\begin{pmatrix}8\-3\end{pmatrix}$. Diagonal matrix scales each component independently.
$\begin{pmatrix}8\-3\end{pmatrix}$. Diagonal matrix scales each component independently.
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What is $\begin{pmatrix}1&-2\3&0\end{pmatrix}\begin{pmatrix}5\1\end{pmatrix}$?
What is $\begin{pmatrix}1&-2\3&0\end{pmatrix}\begin{pmatrix}5\1\end{pmatrix}$?
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$\begin{pmatrix}3\15\end{pmatrix}$. First row: $1(5)+(-2)(1)=3$; second row: $3(5)+0(1)=15$.
$\begin{pmatrix}3\15\end{pmatrix}$. First row: $1(5)+(-2)(1)=3$; second row: $3(5)+0(1)=15$.
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What is the image of $\vec{e}_1=\begin{pmatrix}1\0\end{pmatrix}$ under $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
What is the image of $\vec{e}_1=\begin{pmatrix}1\0\end{pmatrix}$ under $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
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$A\vec{e}_1=\begin{pmatrix}a\c\end{pmatrix}$ (the first column of $A$). Multiplying by $\vec{e}_1$ extracts the first column.
$A\vec{e}_1=\begin{pmatrix}a\c\end{pmatrix}$ (the first column of $A$). Multiplying by $\vec{e}_1$ extracts the first column.
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What is the image of $\vec{e}_2=\begin{pmatrix}0\1\end{pmatrix}$ under $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
What is the image of $\vec{e}_2=\begin{pmatrix}0\1\end{pmatrix}$ under $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$?
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$A\vec{e}_2=\begin{pmatrix}b\d\end{pmatrix}$ (the second column of $A$). Multiplying by $\vec{e}_2$ extracts the second column.
$A\vec{e}_2=\begin{pmatrix}b\d\end{pmatrix}$ (the second column of $A$). Multiplying by $\vec{e}_2$ extracts the second column.
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What is $\begin{pmatrix}1&0\0&1\end{pmatrix}\begin{pmatrix}-3\9\end{pmatrix}$?
What is $\begin{pmatrix}1&0\0&1\end{pmatrix}\begin{pmatrix}-3\9\end{pmatrix}$?
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$\begin{pmatrix}-3\9\end{pmatrix}$. Identity matrix leaves vectors unchanged.
$\begin{pmatrix}-3\9\end{pmatrix}$. Identity matrix leaves vectors unchanged.
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What is $\begin{pmatrix}-1&0\0&1\end{pmatrix}\begin{pmatrix}6\-2\end{pmatrix}$?
What is $\begin{pmatrix}-1&0\0&1\end{pmatrix}\begin{pmatrix}6\-2\end{pmatrix}$?
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$\begin{pmatrix}-6\-2\end{pmatrix}$. Reflects $x$-component across origin, keeps $y$ unchanged.
$\begin{pmatrix}-6\-2\end{pmatrix}$. Reflects $x$-component across origin, keeps $y$ unchanged.
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What is the standard matrix for the transformation $T(x,y)=(ax+by,cx+dy)$?
What is the standard matrix for the transformation $T(x,y)=(ax+by,cx+dy)$?
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$\begin{pmatrix}a&b\c&d\end{pmatrix}$. Coefficients form columns: $(a,c)$ and $(b,d)$.
$\begin{pmatrix}a&b\c&d\end{pmatrix}$. Coefficients form columns: $(a,c)$ and $(b,d)$.
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What is the definition of a matrix transformation $T$ associated with a matrix $A$ acting on vectors?
What is the definition of a matrix transformation $T$ associated with a matrix $A$ acting on vectors?
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$T(\vec{v})=A\vec{v}$ for each vector $\vec{v}$. Matrix transforms vectors by multiplication.
$T(\vec{v})=A\vec{v}$ for each vector $\vec{v}$. Matrix transforms vectors by multiplication.
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What is $\begin{pmatrix}3&0\0&3\end{pmatrix}\begin{pmatrix}-1\4\end{pmatrix}$?
What is $\begin{pmatrix}3&0\0&3\end{pmatrix}\begin{pmatrix}-1\4\end{pmatrix}$?
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$\begin{pmatrix}-3\12\end{pmatrix}$. Scalar matrix multiplies all components by 3.
$\begin{pmatrix}-3\12\end{pmatrix}$. Scalar matrix multiplies all components by 3.
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