به انگلیسی

Spaces that behave like arrows

Mathematics · Linear Algebra · ages 18-19

Name ______________________ Date ____________

With the usual addition and scaling, which set is a vector space whose elements are not arrows?
- All 2 by 2 matrices with a 1 in the top left corner
- All 2 by 2 matrices
- All polynomials with constant term 5
- All arrows in the plane of length 1
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
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
Sam says the solutions of A times x equals b, with b nonzero, also form a subspace. Is Sam right?
Circle one: True False
Why do the solutions of a homogeneous system always form a subspace?
- Every homogeneous system has exactly one solution, the zero vector
- Zero is the only solution of any system
- Linearity keeps sums and scalings of solutions as solutions
Consider the polynomials of degree at most 2 whose value at x = 1 is 0. Do they form a vector space?
- No: adding two such polynomials can push the degree above 2
- No: the zero polynomial has no degree, so it cannot be a member
- Yes: sums and scalings still give 0 at x = 1, and so does zero
Which subset of the plane is a subspace?
- The line y = 3x, all points (x, 3x)
- The line y = 3x + 2
- The circle of radius 1 around the origin
- The points whose coordinates are both whole numbers
Why do all 2 by 2 symmetric matrices form a vector space, while the invertible ones do not?
- Symmetric matrices are arrows in the plane, while invertible matrices are not arrows
- Invertible matrices are too large to form a space
- Symmetry survives sums and scalings; invertibility can break under addition
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
Sam says the polynomials of degree exactly 3 form a vector space. Is Sam right?
Circle one: True False

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

All 2 by 2 matrices · Sums and scalings of 2 by 2 matrices are 2 by 2 matrices, and the zero matrix is included. The other sets miss zero, and the last one holds arrows.
All polynomials of degree at most 3 · Sums and scalings of polynomials of degree at most 3 still have degree at most 3, and none of them is an arrow.
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.
False · Zero is missing: A times zero is zero, never the nonzero b.
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.
Yes: sums and scalings still give 0 at x = 1, and so does zero · If p(1) = 0 and q(1) = 0, then (p + q)(1) = 0 and (cp)(1) = 0. Degrees stay at most 2, and the zero polynomial qualifies.
The line y = 3x, all points (x, 3x) · The line y = 3x holds zero and keeps sums and scalings on the line. The shifted line misses zero, the circle misses zero, and whole-number points fail scaling by 0.5.
Symmetry survives sums and scalings; invertibility can break under addition · Symmetry survives addition and scaling, but a matrix plus its own negative is the singular zero matrix.
It fails scaling: minus 1 times (1, 1) leaves it · Addition holds, but scaling by a negative number escapes the quadrant.
False · The zero polynomial is missing and sums can drop in degree, so closure fails.