Practice question · Put in order
Order these subspaces of ordinary three-dimensional space by increasing dimension.
- The span of (1,0,0), (0,1,0) and (0,0,1)
- The span of (1, 2, 2)
- The subspace containing only the origin
- The plane x + y + z = 0
Hints
- Count the free parameters needed to describe a general member of each set.
- One linear condition on three coordinates leaves two free.
Show the answer
- The subspace containing only the origin
- The span of (1, 2, 2)
- The plane x + y + z = 0
- The span of (1,0,0), (0,1,0) and (0,0,1)
Why
The dimensions are 0, 1, 2 and 3. The plane x + y + z = 0 is cut out by one equation, so two of the three coordinates stay free, each independent linear condition costs exactly one dimension.
Practise Basis and Dimension
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