# [Linear Algebra] problem with finding base and dimension

• Jan 8th 2009, 05:55 AM
gordorin
[Linear Algebra] problem with finding base and dimension
Hi,
I stuck with this question:
U,W are sub spaces of R4[x]:
ImageShack - Image Hosting :: 84809558ns8.jpg
Find base and dimension for:
\$\displaystyle U,W,U+W, \$
and : U intersection B

Thanks a lot!
• Jan 8th 2009, 05:40 PM
HallsofIvy
U is the subspace spanned by \$\displaystyle x^3+ 4x^2- x+ 3\$, \$\displaystyle x^3+ 5x^2+ 5\$, and \$\displaystyle 3x^3+ 10x^2+ 5\$. If those three vectors are independent then they form a basis for U themselves.
Suppose [tex]a(x^3+ 4x^2- x+ 3)+ b(x^3+ 5x^2+ 5)+ c(x^3+ 10x^2+ 5)= 0[/itex]. That gives \$\displaystyle (a+ b+ c)x^3+ (4a+ 5b+ 10c)x^2+ (-a)x+ (3a+ 5b+ 5c)= 0\$. Since \$\displaystyle x^3\$, \$\displaystyle x^2\$, \$\displaystyle x\$, and 1 are independent themselves, we must have a+ b+ c= 0, 4a+ 6b+ 10c= 0, -a= 0, and 3a+ 5b+ 5c= 0. From the third equation, a= 0. so the others are b+ c= 0, 6b+ 10c= 0, and 5b+ 5c= 5(b+ c)= 0. The first and last equations are satisfied as long as c= -b but that makes the other equation 6b+ 10(-b)= -4b= 0 so b= 0 and c= 0. Yes, those three vectors are independent and so form a basis for U which is three dimensional.

Do the same thing with V.