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Undergraduate practice chapters

1. Undergraduate · Chapter 1

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

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2. Undergraduate · Chapter 2

   Diagonal eigenmodes · First eigenvalue a=-2; 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=-2; Second eigenvalue b=-2.

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3. Undergraduate · Chapter 3

   Matrix conditioning · Small diagonal entry a=0.05; 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=0.05; Final data error b=0.1.

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4. Undergraduate · Chapter 4

   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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5. Undergraduate · Chapter 5

   Euler stability · Decay rate a=0.5; Step size h=0.1

   Use n=6 updates, corresponding to time t=nh=0.6. 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=0.1.

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6. Undergraduate · Chapter 6

   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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7. Undergraduate · Chapter 7

   Diagonal eigenmodes · First eigenvalue a=-1.5; 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=-1.5; Second eigenvalue b=-1.5.

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8. Undergraduate · Chapter 8

   Matrix conditioning · Small diagonal entry a=0.1; 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=0.1; Final data error b=0.2.

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9. Undergraduate · Chapter 9

   Logistic differential equations · Capacity K=1.5; Growth rate r=0.4

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=1.5; Growth rate r=0.4.

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10. Undergraduate · Chapter 10

    Harmonic oscillators · Angular frequency ω=1; 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=1; Initial displacement b=-1.5.

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11. Undergraduate · Chapter 11

    Euler stability · Decay rate a=0.75; Step size h=0.2

    Use n=6 updates, corresponding to time t=nh=1.2. 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=0.2.

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12. Undergraduate · Chapter 12

    Two-state Markov chains · Transition 1 → 2: p=0.2; Transition 2 → 1: q=0.2

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.2; Transition 2 → 1: q=0.2.

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13. Undergraduate · Chapter 13

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

    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=-2.

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14. Undergraduate · Chapter 14

    Matrix conditioning · Small diagonal entry a=0.15; Final data error b=0.3

    Use the current perturbation δ=0.18, so Δb=(0,0.18). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.15; Final data error b=0.3.

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15. Undergraduate · Chapter 15

    Directional derivatives · x² coefficient a=1.5; y² coefficient b=1.5

    Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point. Givens: x² coefficient a=1.5; y² coefficient b=1.5.

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16. Undergraduate · Chapter 16

    Harmonic oscillators · Angular frequency ω=1.5; Initial displacement b=-1

    Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=1.5; Initial displacement b=-1.

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17. Undergraduate · Chapter 17

    Euler stability · Decay rate a=1; Step size h=0.3

    Use n=6 updates, corresponding to time t=nh=1.8. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1; Step size h=0.3.

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18. Undergraduate · Chapter 18

    Newton root iteration · Number whose root is sought A=3; Positive starting guess b=1.5

    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=3; Positive starting guess b=1.5.

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19. Undergraduate · Chapter 19

    Uniform convolution · First uniform width a=1.5; Second uniform width b=1.5

    Evaluate at sum value s=1.8, three fifths of the support length. Find the density f_S(s), cumulative probability P(S≤s), and expected sum. Givens: First uniform width a=1.5; Second uniform width b=1.5.

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20. Undergraduate · Chapter 20

    Orthogonal projections · Vector horizontal a=-1.5; Vector vertical b=-1.5

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=-1.5; Vector vertical b=-1.5.

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