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 Linear Algebra Linear Algebra Math Forum

 October 18th, 2017, 06:27 AM #1 Senior Member   Joined: Jan 2016 From: Blackpool Posts: 103 Thanks: 2 bases of the reals^2 question: find 3 different bases of R^2 My answer: I thought for R^2 there are only two standard base vectors e1,e2 where e1=(0,1) e2=(1,0) which spans every vector in R^2 and can be written as a unique linear combination. So what is the 3rd bases of R^2 or could the question be wrong?? Thanks  October 18th, 2017, 06:45 AM #2 Member   Joined: Aug 2011 From: Nouakchott, Mauritania Posts: 85 Thanks: 14 Math Focus: Algebra, Cryptography Hi ! A basis of $\displaystyle {\mathbb R}^2$ is a set of two vectors that span all the vectors of $\displaystyle {\mathbb R}^2$. The famous basis is what you mentioned, which is called the canonical basis $\displaystyle B=(e_1,e_2)$ where $\displaystyle e_1=(1,0)$ and $\displaystyle e_2=(0,1)$. But this is only a single basis - although it has two vectors. Did you get it ? Now for answering your question, we can use this property: Let $\displaystyle w=(w_1,w_2)$ and $\displaystyle v=(v_1,v_2)$, then $\displaystyle (w,v)$ is a basis of $\displaystyle {\mathbb R}^2$ iff the determinant of $\displaystyle (w,v)$ is not null. That means: $\displaystyle \begin{vmatrix} w_1 &v_1 \\ w_2 &v_2 \end{vmatrix}\ne 0$. So, all that you need is finding 3 pairs of vectors that satisfy that condition. Sorry for bad English! Thanks from Jaket1 Last edited by skipjack; October 19th, 2017 at 03:13 PM. October 18th, 2017, 08:36 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,614 Thanks: 2603 Math Focus: Mainly analysis and algebra Note that finding such vectors is straightforward in $\mathbb R^2$. Any pair of non-collinear vectors will do. Thanks from topsquark and Jaket1 Last edited by skipjack; October 19th, 2017 at 03:15 PM. October 18th, 2017, 11:13 AM #4 Senior Member   Joined: Jan 2016 From: Blackpool Posts: 103 Thanks: 2 Ahhh, I see. I didn't realise that b=(e1,e2) only forms one single basis in R^2, so would the second basis vectors be, for example, w=(1,1) v=(1,-1) and the third bases vectors be w=(2,2) v=(2,-2) ?? Would this work since the det(w*v) is not equal to 0? Thanks again! Last edited by skipjack; October 19th, 2017 at 03:16 PM. October 18th, 2017, 11:48 AM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,614 Thanks: 2603 Math Focus: Mainly analysis and algebra Yes, both are bases, but your teacher might not like you quoting those two pairs as they are just scalar multiples of each other. I'd find a third pair. Not that your pair don't have to be at right-angles, just not collinear. Thanks from topsquark and Jaket1 Last edited by skipjack; October 19th, 2017 at 03:16 PM. October 18th, 2017, 07:15 PM #6 Senior Member   Joined: Sep 2016 From: USA Posts: 578 Thanks: 345 Math Focus: Dynamical systems, analytic function theory, numerics By definition, a basis for a space is any spanning set of vectors which is linearly independent. I would also hesitate to ${\bf define}$ a basis in terms of determinants. The "right" way to look for a basis is to iteratively build a set of vectors by adding vectors to your set which don't lie in the span of the current vectors in your set. The problem with thinking of things in terms of determinants is 1. It is not a useful computational tool. 2. It does not give any insight into what a basis is. 3. Determinants only make sense for square matrices. However, non-square matrices show up often in math and questions about their important subspaces (kernel and image) are common. In this case, determinants aren't even defined, but the concept of a basis is. For example, the following 3 vectors $(1,1,1,0), (1,1,0,1),(1,0,1,1)$ span a 3-dimensional subspace of $\mathbb{R}^4$, hence they are a basis for this subspace. But you can't compute their determinant. Thanks from topsquark, Jaket1 and Joppy Last edited by skipjack; October 19th, 2017 at 03:18 PM. Tags bases, reals2 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 AplanisTophet Number Theory 30 June 21st, 2017 09:25 AM Lalitha183 Abstract Algebra 3 June 2nd, 2017 10:02 PM Magnitude Real Analysis 2 February 27th, 2017 08:37 AM mathbalarka Number Theory 1 May 9th, 2013 05:51 AM cernlife Real Analysis 5 May 30th, 2011 08:37 PM

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