Exercise-14

LADR

Exercise 1

Suppose

\(U = \{(x, -x, 2x) \in \mathbb{F}^{3}\ :\ x \in \mathbb{F}\}\) and \(W = \{(x, x, 2x) \in \mathbb{F}^{3}\ :\ x \in \mathbb{F}\}\)

Describe \(U + W\) using symbols, and also give a description of \(U + W\) that uses no symbols.

Solution

Step-1

We have: \[ U + W = \{u + w\ |\ u \in U, w \in W\} \]

Step-2

Let \(u = (x, -x, 2x)\) and \(v = (y, y, 2y)\). Then: \[ U + W = \{(x + y, y - x, 2(x + y)) \in \mathbb{F}^{3}\ |\ x, y \in \mathbb{F}\} \]

Step-3

\(U + W\) is a subspace each of whose elements can be written as the sum of two vectors, one of which is in \(U\) and the other in \(W\). It is the smallest subspace that contains both \(U\) and \(W\).

Note

If \(\mathbb{F} = \mathbb{R}\), then we can give a geometric description of \(U\), \(W\) and \(U + W\).

  • \(U\) is the line passing through the origin and the point \((1, -1, 2)\).
  • \(W\) is the line passing through the origin and the point \((1, 1, 2)\).
  • \(U + W\) is the plane passing through the origin that contains the vectors \((1, -1, 2)\) and \((1, 1, 2)\).

Footnotes

  1. Exercise-14, Exercises 1C, Page-25, Linear Algebra Done Right, Fourth Edition, Sheldon Axler↩︎