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Undergraduate practice chapters

1. Undergraduate · Chapter 381

   Diagonal eigenmodes · First eigenvalue a=0; Second eigenvalue b=2

   Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=0; Second eigenvalue b=2.

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2. Undergraduate · Chapter 382

   Matrix conditioning · Small diagonal entry a=0.4; Final data error b=1

   Use the current perturbation δ=0.6, so Δb=(0,0.6). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.4; Final data error b=1.

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3. Undergraduate · Chapter 383

   Logistic differential equations · Capacity K=2.5; Growth rate r=1.6

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=2.5; Growth rate r=1.6.

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4. Undergraduate · Chapter 384

   Harmonic oscillators · Angular frequency ω=3; Initial displacement b=-0.5

   Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=3; Initial displacement b=-0.5.

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5. Undergraduate · Chapter 385

   Euler stability · Decay rate a=1.75; Step size h=1.2

   Use n=6 updates, corresponding to time t=nh=7.2. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1.75; Step size h=1.2.

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6. Undergraduate · Chapter 386

   Newton root iteration · Number whose root is sought A=2; Positive starting guess b=2.5

   Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=2; Positive starting guess b=2.5.

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7. Undergraduate · Chapter 387

   Two-state Markov chains · Transition 1 → 2: p=0.3; Transition 2 → 1: q=0.8

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.3; Transition 2 → 1: q=0.8.

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8. Undergraduate · Chapter 388

   Orthogonal projections · Vector horizontal a=2; Vector vertical b=-3

   Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=2; Vector vertical b=-3.

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9. Undergraduate · Chapter 389

   Diagonal eigenmodes · First eigenvalue a=0.5; Second eigenvalue b=2.5

   Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=0.5; Second eigenvalue b=2.5.

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10. Undergraduate · Chapter 390

    Matrix conditioning · Small diagonal entry a=0.45; Final data error b=0.1

    Use the current perturbation δ=0.06, so Δb=(0,0.06). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.45; Final data error b=0.1.

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11. Undergraduate · Chapter 391

    Logistic differential equations · Capacity K=3; Growth rate r=1.8

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=3; Growth rate r=1.8.

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12. Undergraduate · Chapter 392

    Harmonic oscillators · Angular frequency ω=0.5; Initial displacement b=2

    Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=0.5; Initial displacement b=2.

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13. Undergraduate · Chapter 393

    Euler stability · Decay rate a=2; Step size h=1.3

    Use n=6 updates, corresponding to time t=nh=7.8. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=2; Step size h=1.3.

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14. Undergraduate · Chapter 394

    Newton root iteration · Number whose root is sought A=3; Positive starting guess b=3

    Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=3; Positive starting guess b=3.

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15. Undergraduate · Chapter 395

    Two-state Markov chains · Transition 1 → 2: p=0.4; Transition 2 → 1: q=0.9

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.4; Transition 2 → 1: q=0.9.

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16. Undergraduate · Chapter 396

    Orthogonal projections · Vector horizontal a=2.5; Vector vertical b=-2.5

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=2.5; Vector vertical b=-2.5.

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17. Undergraduate · Chapter 397

    Diagonal eigenmodes · First eigenvalue a=1; Second eigenvalue b=3

    Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=1; Second eigenvalue b=3.

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18. Undergraduate · Chapter 398

    Matrix conditioning · Small diagonal entry a=0.5; Final data error b=0.2

    Use the current perturbation δ=0.12, so Δb=(0,0.12). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.5; Final data error b=0.2.

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19. Undergraduate · Chapter 399

    Logistic differential equations · Capacity K=3.5; Growth rate r=2

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=3.5; Growth rate r=2.

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20. Undergraduate · Chapter 400

    Euler stability · Decay rate a=0.5; Step size h=0.8

    Use n=6 updates, corresponding to time t=nh=4.8. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=0.5; Step size h=0.8.

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