Precalculus Flashcards: Matrix Transformations Of Vectors

What this deck covers

This deck focuses on Matrix Transformations Of Vectors, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the result of applying $A$ then $B$ to a vector $\vec{v}$, written as one matrix times $\vec{v}$?

Answer: $B(A\vec{v})=(BA)\vec{v}$. Composition of transformations equals product of matrices.

Flashcard 2: What is the output type of multiplying a matrix $A$ by a column vector $\vec{v}$ (with compatible dimensions)?

Answer: A column vector (a matrix of size $m\times 1$). Matrix-vector multiplication preserves the column structure.

Flashcard 3: What is $A\begin{pmatrix}1\\1\end{pmatrix}$ if $A=\begin{pmatrix}2&-1\\4&3\end{pmatrix}$?

Answer: $\begin{pmatrix}1\\7\end{pmatrix}$. $2(1)+(-1)(1)=1$; $4(1)+3(1)=7$.

Flashcard 4: What is $A\vec{e}_2$ for $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ and $\vec{e}_2=\begin{pmatrix}0\\1\end{pmatrix}$?

Answer: $\begin{pmatrix}b\\d\end{pmatrix}$. Multiplying by $\vec{e}_2$ selects second column of $A$.

Flashcard 5: What is $A\vec{e}_1$ for $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ and $\vec{e}_1=\begin{pmatrix}1\\0\end{pmatrix}$?

Answer: $\begin{pmatrix}a\\c\end{pmatrix}$. Multiplying by $\vec{e}_1$ selects first column of $A$.

Flashcard 6: Compute $A\vec{x}$ for $A=\begin{pmatrix}3&0\\0&\frac{1}{2}\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}2\\8\end{pmatrix}$.

Answer: $\begin{pmatrix}6\\4\end{pmatrix}$. $3(2)+0(8)=6$; $0(2)+\frac{1}{2}(8)=4$.

Flashcard 7: What is $\begin{pmatrix}3&0\\0&3\end{pmatrix}\begin{pmatrix}-1\\4\end{pmatrix}$?

Answer: $\begin{pmatrix}-3\\12\end{pmatrix}$. Scalar matrix multiplies all components by 3.

Flashcard 8: What is $\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}8\\-3\end{pmatrix}$?

Answer: $\begin{pmatrix}-3\\8\end{pmatrix}$. Swaps $x$ and $y$ components.

Flashcard 9: What is $\begin{pmatrix}1&k\\0&1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}$ in terms of $x,y,k$?

Answer: $\begin{pmatrix}x+ky\\y\end{pmatrix}$. Shear transformation adds $k$ times $y$ to $x$.

Flashcard 10: What is the standard matrix for the transformation $T(x,y)=(ax+by,cx+dy)$?

Answer: $\begin{pmatrix}a&b\\c&d\end{pmatrix}$. Coefficients form columns: $(a,c)$ and $(b,d)$.

Flashcard 11: What is the $2\times 2$ matrix that swaps coordinates, reflecting across the line $y=x$?

Answer: $\begin{pmatrix}0&1\\1&0\end{pmatrix}$. Swaps $x$ and $y$ coordinates.

Flashcard 12: What is the $2\times 2$ matrix for a counterclockwise rotation by angle $\theta$ in $\mathbb{R}^2$?

Answer: $\begin{pmatrix}\cos\theta&-\sin\theta\\sin\theta&\cos\theta\end{pmatrix}$. Standard rotation matrix using trig functions.

Flashcard 13: What is the $2\times 2$ matrix that reflects points across the $x$-axis in $\mathbb{R}^2$?

Answer: $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$. Negates $y$-coordinate while keeping $x$ unchanged.

Flashcard 14: What is the definition of a matrix transformation $T$ associated with a matrix $A$ acting on vectors?

Answer: $T(\vec{v})=A\vec{v}$ for each vector $\vec{v}$. Matrix transforms vectors by multiplication.

Flashcard 15: Compute $A\vec{x}$ for $A=\begin{pmatrix}1&4\\0&1\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}1\\-2\end{pmatrix}$.

Answer: $\begin{pmatrix}-7\\-2\end{pmatrix}$. $1(1)+4(-2)=-7$; $0(1)+1(-2)=-2$.

Flashcard 16: What is the formula for $A\vec{v}$ when $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ and $\vec{v}=\begin{pmatrix}x\\y\end{pmatrix}$?

Answer: $A\vec{v}=\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}$. Each component is the dot product of a row with the vector.

Flashcard 17: What dimension must a matrix $A$ have to multiply a vector $\vec{v}$ of size $n\times 1$ on the left?

Answer: $A$ must be $m\times n$ so $A\vec{v}$ is $m\times 1$. Matrix cols must match vector rows for multiplication.

Flashcard 18: What is $\begin{pmatrix}-1&0\\0&1\end{pmatrix}\begin{pmatrix}6\\-2\end{pmatrix}$?

Answer: $\begin{pmatrix}-6\\-2\end{pmatrix}$. Reflects $x$-component across origin, keeps $y$ unchanged.

Flashcard 19: What dimension must an $m\times n$ matrix $A$ have to multiply a vector $\vec{x}$ of size $n\times 1$?

Answer: $A$ must be $m\times n$ so $A\vec{x}$ is defined. Number of columns in $A$ must equal rows in $\vec{x}$ for multiplication.

Flashcard 20: What is $\begin{pmatrix}2&0\\0&3\end{pmatrix}\begin{pmatrix}4\\-1\end{pmatrix}$?

Answer: $\begin{pmatrix}8\\-3\end{pmatrix}$. Diagonal matrix scales each component independently.

Flashcard 21: What is $\begin{pmatrix}1&0\\0&1\end{pmatrix}\begin{pmatrix}-3\\9\end{pmatrix}$?

Answer: $\begin{pmatrix}-3\\9\end{pmatrix}$. Identity matrix leaves vectors unchanged.

Flashcard 22: What is the $2\times 2$ matrix for scaling by factor $k$ in both directions in $\mathbb{R}^2$?

Answer: $\begin{pmatrix}k&0\\0&k\end{pmatrix}$. Diagonal matrix with $k$ scales both coordinates by $k$.

Flashcard 23: Compute $A\vec{x}$ for $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}-1\\5\end{pmatrix}$.

Answer: $\begin{pmatrix}9\\17\end{pmatrix}$. $1(-1)+2(5)=9$; $3(-1)+4(5)=17$.

Flashcard 24: What is the $2\times 2$ matrix that reflects points across the $y$-axis in $\mathbb{R}^2$?

Answer: $\begin{pmatrix}-1&0\\0&1\end{pmatrix}$. Negates $x$-coordinate while keeping $y$ unchanged.

Flashcard 25: Identify the dimension error: Can $\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}\begin{pmatrix}7\\8\end{pmatrix}$ be computed?

Answer: No; inner dimensions $3$ and $2$ do not match. Matrix has 3 columns but vector has 2 rows.

Flashcard 26: What is $\begin{pmatrix}1&-2\\3&0\end{pmatrix}\begin{pmatrix}5\\1\end{pmatrix}$?

Answer: $\begin{pmatrix}3\\15\end{pmatrix}$. First row: $1(5)+(-2)(1)=3$; second row: $3(5)+0(1)=15$.

Flashcard 27: What is the dimension of the product $A\vec{x}$ if $A$ is $m\times n$ and $\vec{x}$ is $n\times 1$?

Answer: $A\vec{x}$ is $m\times 1$. Result has rows of $A$ and columns of $\vec{x}$.

Flashcard 28: Identify whether $A\vec{x}$ is defined for $A$ of size $2\times 3$ and $\vec{x}$ of size $2\times 1$.

Answer: Not defined since $3\neq 2$ (inner dimensions do not match). Columns of $A$ (3) must equal rows of $\vec{x}$ (2).

Flashcard 29: What is the image of $\vec{e}_1=\begin{pmatrix}1\\0\end{pmatrix}$ under $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$?

Answer: $A\vec{e}_1=\begin{pmatrix}a\\c\end{pmatrix}$ (the first column of $A$). Multiplying by $\vec{e}_1$ extracts the first column.

Flashcard 30: What is the image of $\vec{e}_2=\begin{pmatrix}0\\1\end{pmatrix}$ under $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$?

Answer: $A\vec{e}_2=\begin{pmatrix}b\\d\end{pmatrix}$ (the second column of $A$). Multiplying by $\vec{e}_2$ extracts the second column.

Flashcard 31: What is the $2\times 2$ matrix for a horizontal shear with factor $k$ (sending $(x,y)$ to $(x+ky,y)$)?

Answer: $\begin{pmatrix}1&k\\0&1\end{pmatrix}$. Adds $k$ times $y$ to $x$, keeping $y$ unchanged.

Flashcard 32: Identify whether $A\vec{x}$ is defined for $A$ of size $3\times 2$ and $\vec{x}$ of size $2\times 1$.

Answer: Defined; the product has size $3\times 1$. Columns of $A$ (2) equals rows of $\vec{x}$ (2).

Flashcard 33: What is $\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}2\\5\end{pmatrix}$?

Answer: $\begin{pmatrix}-5\\2\end{pmatrix}$. Rotates vector 90° counterclockwise.

Flashcard 34: What is the entry formula for $(A\vec{x})_i$ when $A$ is $m\times n$ and $\vec{x}$ is $n\times 1$?

Answer: $(A\vec{x})_i=\sum_{j=1}^{n}a_{ij}x_j$. Row $i$ of $A$ dot product with vector $\vec{x}$.

Flashcard 35: What is the linear-combination form of $A\begin{pmatrix}x\\y\end{pmatrix}$ using columns $\vec{a}_1,\vec{a}_2$ of $A$?

Answer: $A\begin{pmatrix}x\\y\end{pmatrix}=x\vec{a}_1+y\vec{a}_2$. Matrix multiplication is a linear combination of columns.

Flashcard 36: Identify the correct interpretation of $A\vec{x}$ in $\mathbb{R}^n$ when $A$ is $n\times n$.

Answer: $A\vec{x}$ is the image of $\vec{x}$ under the linear transformation $A$. Matrix multiplication represents a linear transformation.

Flashcard 37: Compute $A\vec{x}$ for $A=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}-6\\4\end{pmatrix}$.

Answer: $\begin{pmatrix}-6\\-4\end{pmatrix}$. $1(-6)+0(4)=-6$; $0(-6)+(-1)(4)=-4$.

Flashcard 38: What is $\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}-4\\7\end{pmatrix}$?

Answer: $\begin{pmatrix}-4\\-7\end{pmatrix}$. Reflects $y$-component across origin, keeps $x$ unchanged.

Flashcard 39: Compute $A\vec{x}$ for $A=\begin{pmatrix}2&-1\\0&3\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}4\\-2\end{pmatrix}$.

Answer: $\begin{pmatrix}10\\-6\end{pmatrix}$. $2(4)+(-1)(-2)=10$; $0(4)+3(-2)=-6$.

Flashcard 40: Compute $A\vec{x}$ for $A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ and $\vec{x}=\begin{pmatrix}2\\3\end{pmatrix}$.

Answer: $\begin{pmatrix}-3\\2\end{pmatrix}$. $0(2)+(-1)(3)=-3$; $1(2)+0(3)=2$.