Bounded Linear Operators and the Operator Norm · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

How much can a linear map stretch

Mathematics · Calculus & Analysis · ages 23-24
Name ______________________   Date ____________

  1. What does the operator norm take the supremum over?

    • Only the basis vectors
    • Only the zero vector
    • All unit inputs

  2. On R squared with Euclidean norm, T(x, y) = (3x + 4y, -4x + 3y). What is the operator norm of T?

    • 3
    • 4
    • 5
    • 7

  3. On the real line with absolute value, consider the operator T(x) = 2x. What is its operator norm?

    Answer: ______________

  4. What does one column of a matrix say?

    • Where one basis vector lands
    • The determinant value
    • The full norm

  5. Differentiation on polynomials with the sup norm on [0, 1] is unbounded. Which family of functions shows it?

    • The powers x^n
    • The constant functions
    • The function e^x on [0, 1]
    • Polynomials of degree at most 5

  6. A student says T is bounded exactly when the image of the unit ball is a bounded set. Is that right?

    Circle one:   True   False

  7. The right shift moves each sequence one place right, padding with zero. As an operator on square-summable sequences, what is its norm?

    Answer: ______________

  8. A critic computes stretch in one direction and calls it the norm. What is missing?

    • Nothing; one direction suffices
    • The supremum over unit inputs
    • A bigger matrix

  9. Why does differentiation break every proposed sup-norm bound?

    • Smooth functions are unbounded
    • Derivatives ignore constants
    • Shrinking heights can pair with ever steeper slopes

  10. What is the operator norm of a rotation of the plane, with Euclidean norm?

    • 0
    • 1
    • 2
    • It depends on the angle
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Answer key

For grown-ups. Fold this page away before handing over the rest.

How much can a linear map stretch W1-mt_Qi_mbSNMMx-s1

  1. All unit inputs · The norm is the least upper bound over every direction of length one.
  2. 5 · The rows have length 5 and are orthogonal, so the map is a rotation scaled by 5, and the norm is 5.
  3. 2 · Multiplication by 2 stretches every input by exactly 2, so the operator norm is 2.
  4. Where one basis vector lands · Columns track the basis images that define the map.
  5. The powers x^n · Each power has sup norm 1 but derivative sup norm n, so the powers x^n witness unboundedness.
  6. True · Boundedness of an operator is defined by boundedness of that image.
  7. 1 · Shifting preserves every length, so the norm is exactly 1.
  8. The supremum over unit inputs · One direction can miss bigger stretches elsewhere.
  9. Shrinking heights can pair with ever steeper slopes · Small sup input with huge slope defeats any finite cap.
  10. 1 · Rotations preserve every length, so the norm is 1 regardless of angle.
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