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 uuu_(i=1)^r W_i != V

Alonzo Odom

Alonzo Odom

Answered question

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

Beginner2022-07-21Added 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 ) }

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?