Find a basis for $U= span\{(1,3,2) ,(2,4 ,5) , (5,11,12) , (1,1,3)\}$?

i was trying this question many times, but i could not able to solve this question , i have no any idea or hint to solve this question. I know that all these vector are linearly independent, but i know that all linear independent vector are not basis. I was taking $(1,0,0) , (0,1,0)$ and $(0,0,1)$

If anbody help me i would be very thankful to him thanks in advance

  • $\begingroup$ $(1,3,2) ,(2,4 ,5) , (5,11,12) , (1,1,3)$ are not linearly independent ($4$ vectors, dimension $3$). Try Gaussian elimination to find a basis. $\endgroup$ – mfl Aug 15 '17 at 8:49

Write the matrix with rows the coordinates of the vectors and perform row reduction. The given vectors corresponding to the non-zero rows make up a basis of the span. Alternatively the non-zero rows are the coordinates of the vectors of a basis (these vectors are linear combinations of the given vectors): $$ \begin{bmatrix}1&3&2\\2&4&5\\5&11&12\\1&1&3\end{bmatrix}\rightsquigarrow\begin{bmatrix}1&3&2\\0&-2&1\\0&-4&2\\0&-2&1\end{bmatrix}\rightsquigarrow\begin{bmatrix}1&3&2\\0&-2&1\\0&0&0\\0&0&0\end{bmatrix} $$ Thus, the span of these vectors has dimension $2$, and we've found two bases:

  • $\;\bigl\{(1,3,2), \,(2,4,5)\bigr\}$,
  • $\;\bigl\{(1,3,2), \,(0,-2,1)\bigr\}$.
  • $\begingroup$ thanks@ bernard,,,, $\endgroup$ – lomber Aug 15 '17 at 12:14

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