I was given this question in an assignment asking if $u=(0,2,-2), v=(1,-2,1)$ and $w=(4,2,3)$ will span a line, a plane or all of $\mathbb{R}^3$.
What I have done so far is determined that these vectors are linearly independent by row reducing to calculate a diagonal product that is not equal to $0$, proving that it is not linearly dependent. This means that it is a basis for $\mathbb{R}^3$.
What I am confused about is how do I know whether this will span a plane, a line or $\mathbb{R}^3$.
Does a basis of $\mathbb{R}^3$ imply it spans $\mathbb{R}^3$?
How do I know if a vector spans a plane or a line or $\mathbb{R}^3$?
Best Answer
If you have $3$ linearly independent vectors, they will span a $3$-dimensional space.
If you have $3$ vectors that are linearly dependent, they will span a space of $0$, $1$, or $2$ dimensions.
Your vectors are independent, so they span $\mathbb R^3$.
Here is an example of three vectors that span a plane: $(1,0,0), (0,1,0), (1,1,0)$.
Here is an example of three vectors that span a line: $(1, 1, 0), (2,2,0), (3,3,0)$.
Here is an example of three vectors that span a zero-dimensional space:
$(0,0,0), (0,0,0), (0,0,0)$.