Linear Transformations and Matrices - AP Precalculus

1. What is the image of a linear transformation?

   The set of all output vectors. Range of transformation; all possible output vectors.

2. Find the eigenvalues of matrix $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$.

   $3, 1$. Solve $\det(A - \lambda I) = 0$ for characteristic polynomial.

3. Calculate the trace of $\begin{pmatrix} 7 & 0 \\ 0 & 3 \end{pmatrix}$.

   10. Sum diagonal entries: $7 + 3 = 10$.

4. Identify the linear transformation represented by $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$.

   A scaling by 2 in $x$-direction and 3 in $y$-direction. Diagonal matrix scales each coordinate independently.

5. What is the row-reduced echelon form of $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$?

   $\begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix}$. Second row is multiple of first, so it reduces to zero.

6. Define the kernel of a linear transformation.

   Set of vectors mapped to zero vector by the transformation. Also called null space; vectors that map to zero.

7. State the condition for a linear transformation to be invertible.

   The matrix must be square and have a non-zero determinant. Only square matrices with non-zero determinant are invertible.

8. What is the identity element for matrix addition?

   The zero matrix. Adding zero matrix to any matrix leaves it unchanged.

9. What is the relationship between determinants and area/volume?

   The absolute value of the determinant gives area/volume scale factor. Determinant measures how transformation scales area/volume.

10. What is the definition of an identity matrix?

    A square matrix $I$ with ones on the diagonal and zeros elsewhere. When multiplied by any matrix, returns that same matrix.

11. Identify the matrix representation of $T: \textbf{R}^2 \to \textbf{R}^2$ given $T(1,0) = (3,2)$, $T(0,1) = (1,4)$.

    $\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$. Column 1 is $T(1,0)$ and column 2 is $T(0,1)$.

12. What is the effect of a transformation matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ on a vector?

    Rotates the vector 90 degrees counterclockwise. Standard rotation matrix for 90° counterclockwise.

13. Compute the determinant of $\begin{pmatrix} 6 & 1 \\ 4 & -2 \end{pmatrix}$.

    -14. Apply formula: $(6)(-2) - (1)(4) = -12 - 2 = -14$.

14. What is the effect of a zero determinant on a transformation?

    The transformation collapses the space to a lower dimension. Zero determinant means transformation is not invertible.

15. What is the definition of a linear transformation?

    A function $T: \textbf{R}^n \to \textbf{R}^m$ preserving vector addition and scalar multiplication. Must satisfy $T(u+v) = T(u) + T(v)$ and $T(cv) = cT(v)$.

16. Find the inverse of matrix $\begin{pmatrix} 2 & 0 \\ 0 & -3 \end{pmatrix}$.

    $\begin{pmatrix} \frac{1}{2} & 0 \\ 0 & -\frac{1}{3} \end{pmatrix}$. For diagonal matrix, inverse has reciprocal diagonal entries.

17. What is the condition for two matrices to be equal?

    They must have the same dimensions and equal corresponding entries. Matrix equality requires identical dimensions and entries.

18. How many solutions exist for a system represented by a singular matrix?

    Either zero or infinitely many solutions. Singular matrices lead to dependent equations.

19. What is the standard form of a matrix for a linear transformation?

    Matrix $A$ such that $T(\textbf{x}) = A\textbf{x}$ for all $\textbf{x}$ in $\textbf{R}^n$. Columns of $A$ are images of standard basis vectors.

20. What is the result of multiplying two matrices $A$ and $B$ if they are both $n \times n$ identity matrices?

    An $n \times n$ identity matrix. Identity times identity equals identity.

21. What is the result of any matrix multiplied by the identity matrix?

    The original matrix. Identity matrix is multiplicative identity for matrices.

22. What is the rank of a $3 \times 3$ identity matrix?

    3. Identity matrix has full rank equal to its dimension.

23. What does it mean if a matrix is orthogonal?

    Its transpose is its inverse. Orthogonal matrices preserve lengths and angles.

24. Determine if matrix $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is its own inverse.

    Yes, $A^2 = I$. Check if $A \cdot A = I$; matrix swaps coordinates.

25. Calculate the trace of matrix $\begin{pmatrix} 5 & 2 \\ 3 & 8 \end{pmatrix}$.

    13. Trace is sum of diagonal entries: $5 + 8 = 13$.

26. What is the effect of a matrix transformation with determinant -1?

    It reflects and possibly rotates the space. Negative determinant indicates orientation reversal.

27. Describe a shear transformation in terms of a matrix.

    A transformation that slants the shape of an object. Shear keeps one direction fixed while slanting the other.

28. What property does a diagonal matrix have regarding eigenvectors?

    The unit vectors are its eigenvectors. Diagonal entries are eigenvalues with standard basis eigenvectors.

29. Calculate the determinant of a $3 \times 3$ identity matrix.

    1. Identity matrix always has determinant 1 in any dimension.

30. What is the sum of eigenvalues for a matrix $A$?

    Equal to the trace of matrix $A$. Fundamental property relating eigenvalues to trace.

31. What is the transpose of matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$?

    $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$. Swap rows and columns: $(A^T)_{ij} = A_{ji}$.

32. State the associative property for matrix multiplication.

    $(AB)C = A(BC)$ for matrices $A$, $B$, and $C$. Matrix multiplication is associative but not commutative.

33. What does it mean if a matrix is singular?

    Its determinant is zero. Singular matrices are not invertible and collapse dimensions.

34. What is the result when a linear transformation is applied twice?

    The square of the transformation matrix applied to a vector. Composition of transformations corresponds to matrix multiplication.

35. Identify the matrix product $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$.

    $\begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix}$. Multiply matrices using row-by-column dot products.

36. Find the inverse of the matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.

    $\begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix}$. Use formula $A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.

37. What is the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

    $ad - bc$. Formula for $2 \times 2$ matrix determinant.

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