Matrix Transformations of Vectors - Precalculus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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