# Spanning of Vectors on the Plane

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.

• I can't figure out what are you asking? What is "brief" here? – XRFXLP Jul 3 '19 at 4:22
• Where is " brief " here? – Narasimham Jul 3 '19 at 4:43
• I have edited it – 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).