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Undergraduate practice chapters

1. Undergraduate · Chapter 281

   Diagonal eigenmodes · First eigenvalue a=-1; Second eigenvalue b=0.5

   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=-1; Second eigenvalue b=0.5.

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2. Undergraduate · Chapter 282

   Matrix conditioning · Small diagonal entry a=0.75; Final data error b=0.6

   Use the current perturbation δ=0.36, so Δb=(0,0.36). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.75; Final data error b=0.6.

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3. Undergraduate · Chapter 283

   Logistic differential equations · Capacity K=1.5; Growth rate r=1

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=1.5; Growth rate r=1.

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4. Undergraduate · Chapter 284

   Harmonic oscillators · Angular frequency ω=2.5; 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=2.5; Initial displacement b=-2.

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5. Undergraduate · Chapter 285

   Euler stability · Decay rate a=2; Step size h=1.1

   Use n=6 updates, corresponding to time t=nh=6.6. 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=1.1.

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6. Undergraduate · Chapter 286

   Newton root iteration · Number whose root is sought A=5; Positive starting guess b=3.5

   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=5; Positive starting guess b=3.5.

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7. Undergraduate · Chapter 287

   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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8. Undergraduate · Chapter 288

   Diagonal eigenmodes · First eigenvalue a=-0.5; Second eigenvalue b=1

   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=1.

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9. Undergraduate · Chapter 289

   Matrix conditioning · Small diagonal entry a=0.8; Final data error b=0.7

   Use the current perturbation δ=0.42, so Δb=(0,0.42). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.8; Final data error b=0.7.

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10. Undergraduate · Chapter 290

    Directional derivatives · x² coefficient a=3; y² coefficient b=2.5

    Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point. Givens: x² coefficient a=3; y² coefficient b=2.5.

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11. Undergraduate · Chapter 291

    Logistic differential equations · Capacity K=2; Growth rate r=1.2

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=2; Growth rate r=1.2.

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12. Undergraduate · Chapter 292

    Harmonic oscillators · Angular frequency ω=3; Initial displacement b=-1.5

    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=-1.5.

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13. Undergraduate · Chapter 293

    Euler stability · Decay rate a=0.5; Step size h=0.6

    Use n=6 updates, corresponding to time t=nh=3.6. 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=0.6.

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14. Undergraduate · Chapter 294

    Newton root iteration · Number whose root is sought A=6; Positive starting guess b=4

    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=6; Positive starting guess b=4.

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15. Undergraduate · Chapter 295

    Two-state Markov chains · Transition 1 → 2: p=0.9; Transition 2 → 1: q=0.3

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.9; Transition 2 → 1: q=0.3.

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16. Undergraduate · Chapter 296

    Orthogonal projections · Vector horizontal a=2.5; 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=2.5; Vector vertical b=-3.

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17. Undergraduate · Chapter 297

    Diagonal eigenmodes · First eigenvalue a=0; Second eigenvalue b=1.5

    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; Second eigenvalue b=1.5.

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18. Undergraduate · Chapter 298

    Matrix conditioning · Small diagonal entry a=0.85; Final data error b=0.8

    Use the current perturbation δ=0.48, so Δb=(0,0.48). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.85; Final data error b=0.8.

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19. Undergraduate · Chapter 299

    Logistic differential equations · Capacity K=2.5; Growth rate r=1.4

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=2.5; Growth rate r=1.4.

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20. Undergraduate · Chapter 300

    Euler stability · Decay rate a=0.75; Step size h=0.7

    Use n=6 updates, corresponding to time t=nh=4.2. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=0.75; Step size h=0.7.

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