Julia Workshop

Numerical Computing Exercise List 1

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Numerical Computing

Question 1

Matrices

Question 11

Here’s a list of exercises that can help you practice matrix calculations using Julia.

Matrix Calculations in Julia: Exercise List

These exercises should help you get comfortable with matrix calculations in Julia.

Got it! Here are some more introductory exercises with specific examples to help you get started with matrix calculations in Julia:

Introductory Matrix Calculations in Julia: Exercise List with Examples

  1. Creating a Matrix:
    • Create a 2x2 matrix with values [ [1, 2], [3, 4] ].
    A = [1 2; 3 4]
  2. Matrix Addition:
    • Define two 2x2 matrices \[A = [ [1, 2], [3, 4] ]\] and \[B = [ [5, 6], [7, 8] ]\]. Perform matrix addition.
    A = [1 2; 3 4]
B = [5 6; 7 8]
C = A + B
  3. Matrix Subtraction:
    • Define two 2x2 matrices \[A = [ [5, 6], [7, 8] ]\] and \[B = [ [1, 2], [3, 4] ]\]. Perform matrix subtraction.
    A = [5 6; 7 8]
B = [1 2; 3 4]
C = A - B
  4. Matrix Multiplication:
    • Multiply a 2x2 matrix \[A = [ [1, 2], [3, 4] ]\] by another 2x2 matrix \[B = [ [5, 6], [7, 8] ]\].
    A = [1 2; 3 4]
B = [5 6; 7 8]
C = A * B
  5. Matrix Transposition:
    • Transpose a 2x2 matrix \[A = [ [1, 2], [3, 4] ]\].
    A = [1 2; 3 4]
A_transpose = A'
  6. Determinant:
    • Compute the determinant of a 2x2 matrix \[A = [ [1, 2], [3, 4] ]\].
    A = [1 2; 3 4]
det_A = det(A)
  7. Solving Linear Systems:
    • Solve the system of linear equations represented by \[Ax = b\], where \[A = [ [2, 1], [1, 3] ]\] and \[b = [ [8], [13] ]\].
    A = [2 1; 1 3]
b = [8; 13]
x = A \ b

These examples should give you a solid start with matrix calculations in Julia. Let me know if you need more exercises or if you’d like explanations for any of these examples!

  1. Basic Matrix Creation:
    • Create a 3x3 identity matrix.
    • Create a 4x4 zero matrix.
    • Create a 3x3 matrix with random values between 0 and 10.
  2. Matrix Addition and Subtraction:
    • Define two 3x3 matrices and perform matrix addition.
    • Define two 3x3 matrices and perform matrix subtraction.
  3. Matrix Multiplication:
    • Multiply a 3x3 matrix by a 3x1 vector.
    • Perform element-wise multiplication of two 3x3 matrices.
  4. Matrix Transposition:
    • Transpose a given 4x4 matrix and verify its properties.
  5. Determinant and Inverse:
    • Compute the determinant of a 3x3 matrix.
    • Find the inverse of an invertible 3x3 matrix.
    • Verify the result by multiplying the matrix with its inverse.
  6. Eigenvalues and Eigenvectors:
    • Calculate the eigenvalues of a given 2x2 matrix.
    • Compute the eigenvectors corresponding to the eigenvalues.
  7. Solving Linear Systems:
    • Solve a system of linear equations represented by a matrix equation \(Ax = b\), where A is a 3x3 matrix and b is a 3x1 vector.
  8. Matrix Norms:
    • Compute the Frobenius norm and the infinity norm of a 3x3 matrix.
  9. LU Decomposition:
    • Perform LU decomposition of a 3x3 matrix and verify the result.
  10. Singular Value Decomposition (SVD):
    • Perform SVD on a 3x3 matrix and explain the significance of the results.