Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $A=\{a_1, a_2, a_3\}$ be a basis, of which each vector is aligned to a cartesian axis. Given a vector $v_{\langle A\rangle}$, how can I get transform it to the standard, canonical basis so it becomes $v_{\langle I\rangle}$?

I must apologize if this question is silly, I've completely forgotten my linear algebra classes (which makes me want to take them again).

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

First express the basis vectors $\{a_1 , a_2, a_3\}$ in the standard basis: $$ a_{1, \langle I \rangle} = \pmatrix{a_{1,1} \\ a_{1,2} \\ a_{1,3} } \\ a_{2, \langle I \rangle} = \pmatrix{a_{2,1} \\ a_{2,2} \\ a_{2,3} } \\ a_{3, \langle I \rangle} = \pmatrix{a_{3,1} \\ a_{3,2} \\ a_{3,3} } \\$$ Form the matrix $$ A = \pmatrix{a_{1,1} & a_{2,1} & a_{3,1} \\ a_{1,2} & a_{2,2} & a_{3,2} \\ a_{1,3} & a_{2,3} & a_{3,3}} $$ Now, given any $v_{\langle A \rangle} = (v_1 , v_2, v_3),$ we have $$v_{\langle I \rangle} = A \pmatrix{v_1 \\ v_2 \\v_3}. $$


TL;DR $v_{\langle I \rangle}= (v_1, v_2, v_3)_{\langle A \rangle} = v_1 a_{1, \langle I \rangle} + v_2 a_{2, \langle I \rangle} + v_3 a_{3, \langle I \rangle}.$

share|improve this answer
    
I spent way too long messing with the matrix implementation, while the TL;DR was much simpler and faster. Thanks, although now I feel bad and want to take lin. alg. again. –  Lazlo Jul 2 '12 at 17:31
1  
@LazloBonin If you can't enrol in a class, you can go through Gilbert Strang's videos here for example –  user2468 Jul 2 '12 at 17:56
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.