Problems: Evaluate all the vectors that span the set of all vectors on the plane $2x+3y-z=0$. Are all the vectors linearly independent? Give the explanation.

Attempt (source from my friend): This plane is 2-dimensional hence cannot be spanned by a single vector but is spanned by any set $\{p,q\}$ where $p,q\in V$ are independent.

Actually I need more explanation for this problems. Thanks for your help.

  • $\begingroup$ I can't figure out what are you asking? What is "brief" here? $\endgroup$ – XRFXLP Jul 3 '19 at 4:22
  • $\begingroup$ Where is " brief " here? $\endgroup$ – Narasimham Jul 3 '19 at 4:43
  • $\begingroup$ I have edited it $\endgroup$ – Shane Dizzy Sukardy Jul 3 '19 at 5:06

The plane is nothing more than $$\begin{align} \textsf{W} &= \{(x,y,z)\in \mathbb{R}^3:\, 2x+3y-z=0\} \\ &= \{(x,y,z)\in \mathbb{R}^3:\, z=2x+3y\} \\ &= \{(x,y,2x+3y):\, x,y\in \mathbb{R}\} \\ &= \{x(1,0,2)+y(0,1,3):\, x,y\in \mathbb{R}\} \end{align}$$ that is, any vector in the plane can be written as a linear combination of the vectors $(1,0,2)$ and $(0,1,3)$. Now verify that the set formed by these vectors is linearly independent, that is, if $$a_1(1,0,2)+a_2(0,1,3)=(0,0,0)$$ for some scalars $a_1,a_2$, necessarily implies that $a_1=a_2=0$ (which is clear).


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