Linear Algebra Subspaces Basis

[bob258173498]

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

 

[bob258173498]

help please!

 

[bob258173498]

Please guys

 

[Evo]

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.

 

[bob258173498]

V = (v_1, v_2, v_3)
U = ? it never told us

Please can someone help.

 

[bob258173498]

Guys please, I'm begging you

 

[micromass]

Start off by finding a basis for $U\cap W$.

 

[bob258173498]

U = (v_1, v_2)
W = (v_2, v_3)

U∩W = (v_2)

 

[bob258173498]

please i need to this quickly

 

[micromass]

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 $U\cap W$, 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.

 

[Evo]

24 hour time out due to ignoring the rules, ignoring the warning and continuing to bump.

 

[berkeman]

please i need to this quickly

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