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# Vector Subspaces

Module by: Mark A. Davenport. E-mail the author

## Defintition 1

A (non-empty) subset WW of VV is called a supspace of VV if for any x,yWx,yW, span ({x,y})W span ({x,y})W.

Note that this definition easily implies that:

• 0 W 0 W
• WW is itself a vector space

## Example 1: Which of these are subspaces?

•     [No]
•     [Yes]
• V=R5V=R5, W={x:x4=0,x5=0}W={x:x4=0,x5=0} [Yes]
• V=R5V=R5, W={x:x4=1,x5=1}W={x:x4=1,x5=1} [No]
• V=C[0,1]V=C[0,1], W=W={polynomials of degree NN} [Yes]
• V=C[0,1]V=C[0,1], W={f:fW={f:f is bandlimited to BB} [Yes]
• V=RNV=RN, W={x:x hasnomorethan 5 nonzerocomponents ,i.e.,x05}W={x:x hasnomorethan 5 nonzerocomponents ,i.e.,x05} [No]

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