Let k be an infinite field.Let V be a vector space over k and W...

Alonzo Odom

Alonzo Odom

Answered

2022-07-20

Let k be an infinite field.
Let V be a vector space over k and W 1 , . . . , W r proper subspaces of V.
Show that i = 1 r W i V .
I tried the following:
for all j { 1 , . . . , r }, I take w j W j such that w j W i whenever j i, so I know that w 1 + + w r V. If w 1 + + w r i = 1 r W i , then there is l { 1 , . . . , r } such that w 1 + + w r W l . I don't find because w 1 + + w r W l is absurd.
Is this correct reasoning, or is there other way for me to prove this?

Answer & Explanation

Caylee Davenport

Caylee Davenport

Expert

2022-07-21Added 14 answers

The assertion seems to be false. Take k = F 2 and V = k k = { ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) }. Now you can take r=3 and the following proper subspaces of W 1 = { ( 0 , 0 ) , ( 1 , 0 ) }, W 2 = { ( 0 , 0 ) , ( 1 , 1 ) }, and W 3 = { ( 0 , 0 ) , ( 1 , 1 ) }

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