
Linear Algebra Linear Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
December 1st, 2018, 08:05 AM  #1 
Newbie Joined: Dec 2018 From: kazakhstan Posts: 1 Thanks: 0  Linear algebra, sum of subspaces
Let U1, U2, U3 be subspaces of R^4: U1 = {(a,b,c,d):a=b=c} U2 = {(a,b,c,d):a+bc+d=0; c2d=0} U3={(a,b,c,d):3a+d=0} show that: a) U1 + U2 = R^4; b) U2 + U3 = R^4; c) U1 + U3 = R^4; whish of the sums are direct sum? 
December 1st, 2018, 10:58 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,203 Thanks: 1157 
can you write down the basis vectors for each of of the U's? If so for each problem use those to find the basis of the sum and use Gaussian elimination to find the resulting basis. If the equality is true you should end up with a 4x4 Identity matrix. U1 spans $R^2$ U2 spans $R^2$ U3 spans $R^3$ So if I understand the term direct sum it should be pretty obvious which pair of these is the only possible candidate for a direct sum. Last edited by romsek; December 1st, 2018 at 11:52 AM. 

Tags 
algebra, linear, subspaces, sum 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Intersection and Sum of Linear Subspaces in Rn  zylo  Linear Algebra  5  May 24th, 2016 11:20 AM 
about Linear subspaces  Alex010  Linear Algebra  1  May 10th, 2016 11:19 AM 
Linear Transformations in Linear algebra  matqkks  Linear Algebra  1  February 7th, 2012 02:39 PM 
Subspaces and Algebra  Warpenguin  Linear Algebra  1  August 31st, 2011 06:40 AM 
Linear Subspaces...  Babaloo2u  Linear Algebra  1  November 10th, 2010 04:33 AM 