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 June 27th, 2014, 12:17 AM #1 Newbie   Joined: Jun 2014 From: italy Posts: 9 Thanks: 0 base of the orthogonal complement of a subspace In $\displaystyle R^3$ the subspace $\displaystyle V_1$ is generated by the vector $\displaystyle v_1 = (1,2,-1)$ and the subspace $\displaystyle V_2$ is generated by the vector $\displaystyle v_2 = (1,1,-1)$ Find a base of $\displaystyle (V_1 + V_2)^{\bot}$ and a base of $\displaystyle V_1^{\bot}\cap V_2^{\bot}$ $\displaystyle V_1+V_1 = \langle (1,2,-1),(1,1,-1)\rangle$ Let $\displaystyle v= (x,y,z) \in R^3$ then $\displaystyle v \in (V_1 + V_2)^{\bot}$ if \displaystyle \left\{ \begin{aligned} v * (1,2,-1) &=0 \\ v * (1,1,-1) &=0 \ \end{aligned} \right. \displaystyle = \left\{ \begin{aligned} x+2y-z &=0 \\ x+y-z &=0 \ \end{aligned} \right. the solution is: \displaystyle \left\{ \begin{aligned} x &= z \\ y &=0 \ \end{aligned} \right. So $\displaystyle (x,y,z)=(z,0,z)= z(1,0,1)$ Thus $\displaystyle (V_1 + V_2)^{\bot} = \langle(1,0,1)\rangle$ $\displaystyle V_1 = \langle (1,2,-1)\rangle$ $\displaystyle V_2 = \langle (1,1,-1)\rangle$ So $\displaystyle v \in V_1^{\bot}\cap V_2^{\bot}$ if \displaystyle \left\{ \begin{aligned} v*(1,2,-1) &= 0 \\ v*(1,1,-1) &= 0 \ \end{aligned} \right. So it is the same as before thus $\displaystyle V_1^{\bot}\cap V_2^{\bot} = (V_1 + V_2)^{\bot} = \langle(1,0,1)\rangle$ I would like to know if it is solved in the right way. June 27th, 2014, 06:46 AM #2 Senior Member   Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108 You solved it correctly. This problem is probably supposed to illustrate the fact that $(V_1+V_2)^\perp=V_1^\perp\cap V_2^\perp$ for all $V_1,V_2$. Thanks from david940 Tags base, complement, orthogonal, subspace Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post caters Number Theory 9 May 22nd, 2014 02:46 AM walter r Linear Algebra 5 October 16th, 2013 10:57 AM Snkhtng Linear Algebra 3 December 30th, 2011 01:25 PM Tapas Bose Number Theory 10 August 6th, 2010 12:07 PM momesana Algebra 4 December 3rd, 2009 06:13 PM

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