Bounded Linear Operators and the Operator Norm · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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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. Let T be a linear map between normed spaces. A student says T is bounded exactly when the image of the closed 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 is differentiation unbounded on polynomials with the sup norm on [0, 1]?

    • Every smooth function has infinite sup norm on [0, 1]
    • Differentiation leaves the norm of every polynomial unchanged
    • Unit inputs have unbounded derivative norms

  10. On L2 of [0, 1], T multiplies each f by g(x) = 1 + x. What is the operator norm of T?

    • 1.5
    • 2
    • 1
    • 3
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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 · Each column is the image of a basis vector, and these images 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 · True. A bound on T also bounds the image of the unit ball. Conversely, a bound M on that image scales by linearity to ||Tx|| <= M||x|| for every x.
  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. Unit inputs have unbounded derivative norms · The polynomials x^n have sup norm 1 while their derivatives have sup norm n. No finite bound works for all n.
  10. 2 · |g| is largest at x = 1, where it equals 2, so the norm of multiplication by g is 2.
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