Transformation of Coordinates; Matrices
Exercise 1 : Shift Paraboloid
The vertex of the paraboloid shown in Fig. 14.18 is at a distance 2 from the origin of the coordinates. The equation is $z = 2 + x^2 + y^2$. What is the transformation which will shift the paraboloid so that its vertex coincides with the origin O?
Exercise 2 : Line Under Translation
The equation of a certain straight line is $y = -3x + 5$. What will its equation be in a new $x’$-$y’$ coordinate system due to a shift of the origin of $(-2,3)$?
Exercise 3 : Rotation (2D)
A two-dimensional system of coordinates is rotated through an angle of $\pi/3$. The transformation matrix is:
$$ \begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix} $$What is the new vector $\mathbf{r}’$ if $\mathbf{r} = (2,4)$?
Exercise 4 : Rotated Line Equation
Given the equation $y = -x/\sqrt{3} + 2$, if the system of coordinates is rotated through an angle of $60^\circ$ obtain an expression for the equation in the rotated system.
Exercise 5 : Rotation About z-Axis (3D)
A three-dimensional system of coordinates is rotated about the $z$-axis through an angle of $30^\circ$. Obtain the transformed vector $\mathbf{r}’$ if $\mathbf{r} = (3,3,3)$.
Exercise 6 : Matrix Addition and Subtraction
Given the two matrices $A$ and $B$ where:
$$ A = \begin{pmatrix} 1 & 3 \\ 2 & 5 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 2 & 0 \\ -1 & 3 \end{pmatrix} $$evaluate
- $A + B$,
- $A - B$.
Exercise 7 : Scalar Multiplication and Noncommutativity
Let
$$ A = \begin{pmatrix} 2 & 7 \\ 3 & 0 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 9 & 3 \\ 1 & -2 \end{pmatrix} $$- Evaluate the matrix $6A$.
- Show that the expression $AB \neq BA$.
Exercise 8 : Matrix Multiplication
Given
$$ A = \begin{pmatrix} 1 & 2 \\ 7 & 3 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} -1 & 0 \\ 2 & 3 \end{pmatrix} $$evaluate $AB$.
Exercise 9 : Matrix-Vector Product
Evaluate the product $A \mathbf{r}$ if
$$ A = \begin{pmatrix} 1 & -2 \\ 5 & 7 \end{pmatrix}, \quad \mathbf{r} = \begin{pmatrix} x \\ y \end{pmatrix} $$Exercise 10 : Transpose Properties
Given
$$ A = \begin{pmatrix} 1 & 2 \\ 4 & -3 \\ 3 & 0 \end{pmatrix} $$evaluate
- $A^T$
- $(A^T)^T$
Exercise 11 : Skew-Symmetric Matrix
How many independent entries are there in a skew-symmetric $3 \times 3$ matrix?
Exercise 12 : Skew-Symmetric Matrix Decomposition
Decompose into a symmetric and a skew-symmetric matrix:
$$ \begin{pmatrix} 54 & 1 & 1 \\ 0 & 26 & 20 \\ 8 & 84 & 9 \end{pmatrix} $$Exercise 13 : Inverse Matrix
If
$$ A = \begin{pmatrix} 1 & 0 & 3 \\ 2 & -3 & 1 \\ 1 & 2 & 2 \end{pmatrix} \quad \text{and} \quad A^{-1} = \frac{1}{13} \begin{pmatrix} -8 & 6 & 9 \\ -3 & -1 & 5 \\ 7 & -2 & -3 \end{pmatrix} $$show that $A A^{-1} = A^{-1} A = I$.
Exercise 14 : Determinant and Volume
Compute the determinant of
$$ B = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 3 & 4 \\ 0 & -1 & 2 \end{pmatrix} $$and interpret the absolute value of $\det(B)$ as the volume scaling factor of the linear map in $\mathbb{R}^3$.
Exercise 15 : Rank and Linear Independence
Find the rank of
$$ C = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 0 & 1 \end{pmatrix} $$and determine whether its columns are linearly independent.
Exercise 16 : Eigenvalues and Eigenvectors
Find the eigenvalues and eigenvectors of
$$ D = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} $$Compute the characteristic polynomial and diagonalize if possible.
Exercise 17 : Diagonalization
Determine whether
$$ E = \begin{pmatrix} 3 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{pmatrix} $$is diagonalizable. If so, find an invertible $P$ and diagonal $\Lambda$ such that $E = P \Lambda P^{-1}$.
Exercise 18 : Orthogonal Projection
Find the matrix that projects vectors in $\mathbb{R}^3$ orthogonally onto the line spanned by $\mathbf{u} = (1,2,2)$.
Exercise 19 : Gram-Schmidt Orthonormalization
Apply the Gram–Schmidt process to the vectors $\mathbf{v}_1 = (1,1,0)$, $\mathbf{v}_2 = (1,0,1)$ to produce an orthonormal set.
Exercise 20 : Change of Basis
Let $\mathcal{B} = {(1,1),(1,-1)}$ be a basis of $\mathbb{R}^2$. Find the change-of-basis matrix from $\mathcal{B}$-coordinates to standard coordinates, and compute the coordinates of $(3,1)$ in the $\mathcal{B}$ basis.
Exercise 21 : Composition of Transformations
Consider a rotation in the plane by $45^\circ$ followed by a reflection in the $x$-axis. Write down the matrices for each transformation, compute their product, and describe the resulting transformation.