Matrices

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 $(-2,3)$? ...

Eigenvalues and Eigenvectors of Real Matrices Exercise 1 : Eigenvalues — 2x2 real matrix For $A = \begin{pmatrix} 4 & 2 \ 1 & 3 \end{pmatrix}$ find the eigenvalues. Exercise 2 : Real vs Complex Eigenvalues — 2x2 question Is it possible for a real $2 \times 2$ matrix to have one real and one complex eigenvalue? Exercise 3 : No Real Eigenvalues — rotation-like matrix Prove that there are no real eigenvalues of the matrix $A = \begin{pmatrix} 3 & 2 \ -2 & 1 \end{pmatrix}$ ...

Vectors and Matrices in Python Solutions Exercise 1 : Vectors in $\mathbb{R}^7$ Consider the following vectors: $$ u = (0.5, 0.4, 0.4, 0.5, 0.1, 0.4, 0.1), \quad v = (-1, -2, 1, -2, 3, 1, -5) $$Using Python and NumPy: Check whether $u$ and $v$ are unit vectors. Compute the dot product of $u$ and $v$. Determine if $u$ and $v$ are orthogonal. import numpy as np u = np.array([0.5, 0.4, 0.4, 0.5, 0.1, 0.4, 0.1]) v = np.array([-1, -2, 1, -2, 3, 1, -5]) # 1. Check if u and v are unit vectors norm_u = np.linalg.norm(u) norm_v = np.linalg.norm(v) print(norm_u, norm_v) # 2. Dot product dot_uv = np.dot(u, v) print(dot_uv) # 3. Orthogonality print(np.isclose(dot_uv, 0)) Exercise 2 : Norms and Orthogonality Consider the following vectors in $\mathbb{R}^9$: ...

Vectors and Matrices in Python In this worksheet, you will use Python (NumPy) to perform vector and matrix operations. For each exercise, write Python code to compute the required results and verify them numerically. Exercise 1 : Vectors in $\mathbb{R}^7$ Consider the following vectors: $$ u = (0.5, 0.4, 0.4, 0.5, 0.1, 0.4, 0.1), \quad v = (-1, -2, 1, -2, 3, 1, -5) $$Using Python and NumPy: Check whether $u$ and $v$ are unit vectors. Compute the dot product of $u$ and $v$. Determine if $u$ and $v$ are orthogonal. Exercise 2 : Norms and Orthogonality Consider the following vectors in $\mathbb{R}^9$: ...

Vectors and Matrices Exercise 1 Consider the following vectors in $ \mathbb{R}^7 $: $$ u = (0.5, 0.4, 0.4, 0.5, 0.1, 0.4, 0.1), \quad v = (-1, -2, 1, -2, 3, 1, -5) $$ Check if $ u $ and $ v $ are unit vectors. Calculate the dot product of the vectors $ u $ and $ v $. Are $ u $ and $ v $ orthogonal? Exercise 2 Consider the following vectors in $ \mathbb{R}^9 $: ...