Linear Algebra Subspaces Basis

  • #1

Homework Statement



a) If U and W are subspaces of R^3, show that it is possible to find a basis B for R^3 such that one subset of B is a basis for U and another subset of B (possibly overlapping) is a basis for W.

b) If U and W are subspaces of a finite-dimensional vector space V, show that it is possible to find a basis for V such that one subset of that basis is a basis for U and another subset of that basis (possibly overlapping) is a basis for W.


Homework Equations



none really

The Attempt at a Solution



B = ( v_1, v_2, v_3 ), such that these vectors are a basis for V.

Then:

V = t_1*v_1 + t_2*v_2 + t_3*v_3

...stuck here
 

Answers and Replies

  • #3
Please guys
 
  • #4
Evo
Mentor
23,530
3,140

Homework Statement



a) If U and W are subspaces of R^3, show that it is possible to find a basis B for R^3 such that one subset of B is a basis for U and another subset of B (possibly overlapping) is a basis for W.

b) If U and W are subspaces of a finite-dimensional vector space V, show that it is possible to find a basis for V such that one subset of that basis is a basis for U and another subset of that basis (possibly overlapping) is a basis for W.


Homework Equations



none really

The Attempt at a Solution



B = ( v_1, v_2, v_3 ), such that these vectors are a basis for V.

Then:

V = t_1*v_1 + t_2*v_2 + t_3*v_3

...stuck here
People will help if they wish. They probably will not help you since you keep bumping your thread every few minutes, which is against the rules.
 
  • #5
V = (v_1, v_2, v_3)
U = ??? it never told us

Please can someone help.
 
  • #6
Guys please, I'm begging you
 
  • #7
22,129
3,297
Start off by finding a basis for [itex]U\cap W[/itex].
 
  • #8
U = (v_1, v_2)
W = (v_2, v_3)

U∩W = (v_2)
 
  • #9
please i need to this quickly
 
  • #10
22,129
3,297
U = (v_1, v_2)
W = (v_2, v_3)

U∩W = (v_2)

No, what makes you think that U and W are two-dimensional?? U and W can be anything!!

First construct a basis for [itex]U\cap W[/itex], then extend this basis to a basis of U and extend it to a basis of W.


please i need to this quickly

Messages like this actually incline people to help you less. Just so you know.
 
  • #11
Evo
Mentor
23,530
3,140
24 hour time out due to ignoring the rules, ignoring the warning and continuing to bump.
 
  • #12
berkeman
Mentor
60,384
10,678
please i need to this quickly

Do you need it quickly because it is part of a midterm exam?
 

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