Vector Spaces and Subspaces · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Spaces that behave like arrows

Mathematics · Linear Algebra · ages 18-19
Name ______________________   Date ____________

  1. Which collection is a vector space whose elements are not arrows?

    • All polynomials of degree at most 3, with the usual addition and scaling
    • The set {1, 2, 3} with ordinary arithmetic
    • All 2 by 2 invertible matrices
    • All arrows in the plane

  2. Which collection is a vector space whose elements are not arrows?

    • The set {1, 2, 3} with ordinary arithmetic
    • All polynomials of degree at most 3
    • All 2 by 2 invertible matrices

  3. Which of these subsets of the plane is a subspace?

    • The x-axis, points of the form (x, 0)
    • The line y equals x plus 1
    • The first quadrant, both coordinates non negative

  4. Sam says the solutions of A times x equals b, with b nonzero, also form a subspace. Is Sam right?

    Circle one:   True   False

  5. Why do the solutions of a homogeneous system always form a subspace?

    • Every homogeneous system has exactly one solution
    • Zero is the only solution of any system
    • Linearity keeps sums and scalings of solutions as solutions

  6. Why do all 2 by 2 symmetric matrices form a vector space, while the invertible 2 by 2 matrices do not?

    • Sums and scalings of symmetric matrices stay symmetric, but sums of invertible matrices can be singular
    • Symmetric matrices are square and invertible ones are not
    • Invertible matrices are too large to form a space
    • Symmetric matrices are arrows

  7. Which of these subsets of the plane is a subspace?

    • The x-axis, all points of the form (x, 0)
    • The line y equal to x plus 1
    • The first quadrant, where both coordinates are non negative
    • The unit circle

  8. Why do all 2 by 2 symmetric matrices form a vector space, while the invertible ones do not?

    • Symmetric matrices are arrows and invertible ones are not
    • Invertible matrices are too large to form a space
    • Sums and scalings of symmetric matrices stay symmetric, but sums of invertible ones can be singular

  9. Zoe claims the first quadrant is a subspace because it holds zero and is closed under addition. What is her mistake?

    • It fails scaling: minus 1 times (1, 1) leaves it
    • It is not closed under addition
    • It misses the zero vector

  10. Sam says the polynomials of degree exactly 3 form a vector space. Is Sam right?

    Circle one:   True   False

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Answer key

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

Spaces that behave like arrows W1-mt_Zrddx-E6_n-s1

  1. All polynomials of degree at most 3, with the usual addition and scaling · Sums and scalings of low degree polynomials stay low degree polynomials, and no element is an arrow.
  2. All polynomials of degree at most 3 · Sums and scalings of low degree polynomials stay low degree, and none is an arrow.
  3. The x-axis, points of the form (x, 0) · The x-axis holds zero and stays closed, while each other candidate fails at least one check.
  4. False · Zero is missing: A times zero is zero, never the nonzero b.
  5. Linearity keeps sums and scalings of solutions as solutions · If A times x and A times y are zero, so are A times their sum and any scaling.
  6. Sums and scalings of symmetric matrices stay symmetric, but sums of invertible matrices can be singular · Symmetry survives addition and scaling, but adding a matrix to its own negative gives the zero matrix, which is singular.
  7. The x-axis, all points of the form (x, 0) · The x-axis holds zero and stays closed under addition and scaling, while each other candidate fails at least one check.
  8. Sums and scalings of symmetric matrices stay symmetric, but sums of invertible ones can be singular · Symmetry survives addition and scaling, but a matrix plus its own negative is the singular zero matrix.
  9. It fails scaling: minus 1 times (1, 1) leaves it · Addition holds, but scaling by a negative number escapes the quadrant.
  10. False · The zero polynomial is missing and sums can drop in degree, so closure fails.
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