Exercises: Subvector Spaces

Problem 1

The subset \(S\) of the plane below consists of a grid of horizontal and vertical points all of which have the same distance from their neighbors. Is \(S\) closed under addition?

Problem 2

The set \(S\) below consists of the rectangle shown below (including its interior). Is it closed under addition?

Problem 3

Is the set \(S\) below consisting of the rectangle below closed under scalar multiplication?

Problem 4

Let \(S\) be the collection of points in the ellipse below. Is \(S\) closed under addition?

Problem 5

Let \(S\) consist of the family of parallel lines below. The distance of any line from its two neighbors is the same. Is \(S\) closed under addition?

Problem 6

For each of the sets of vectors below determine if it is a subvector space of \(\RNrSpc{2}\).

  1. The set \(N\) containing only the zero vector.

  2. The set \(L\) consisting of all multiples of \(\Vect{a} = (-2,0)\).

  3. The set of \(T\) of vectors \(\Vect{x} = (x,y)\) with \(3x-2=2\)

  4. The set \(P\) of vectors \(\Vect{x} = (x,y)\) with \(x^3+y=0\).

  5. The set \(E\) of vectors \(\Vect{x} = (x,y)\) with \(-x+ey=0\)

Problem 7

For each of the sets of vectors below determine if it is a subvector space of \(\RNrSpc{3}\).

  1. The set \(L\) of all vectors \(\Vect{x} = (x,y,z)\) with \(x=1+t\), \(y=2t\), and \(z=-t\), with \(t\in\RNr\) arbitrary.

  2. The set \(Q\) of vectors \(\Vect{x} = (x,y,z)\) in \(\RNrSpc{3}\) with \(x=\pi t\), \(y=2t^2\), and \(z=t\), where \(t\in\RNr\) is arbitrary.

  3. The set \(K\) of vectors \(\Vect{x}=(x,y,z)\) in \(\RNrSpc{3}\) with \(x=2t\), \(y=2t\), and \(z=2\).

  4. The set \(R\) of vectors \(\Vect{x}=(x,y,z)\) in \(\RNrSpc{3}\) with \(x=4t\), \(y=t\), and \(z=-2t\).

  5. The set \(\Pi\) of vectors \(\Vect{x} = (x,y,z)\) in \(\RNrSpc{3}\) with \(z=-x+y\).

  6. The set \(Z\) of vectors \(\Vect{x} = (x,y,z)\) in \(\RNrSpc{3}\) with \(x-y+z=-1\).

Problem 8

If \(V\) and \(W\) are subspaces of \(\RNrSpc{n}\), show that the union of \(V\) and \(W\) need not be a subvector space of \(\RNrSpc{n}\).

Problem 9Subspace of subspace is subspace

For a subspace \(V\) of \(\RNrSpc{n}\), consider a subset \(S\) of \(V\) which contains the \(\Vect{0}\)-vector and is closed under addition and scalar multiplication. Show that \(S\) is a subspace of \(\RNrSpc{n}\).