I am given $3$ vectors that are linearly independent. I am trying to figure our if they span all of $\mathbb{R}^3$ to declare them as basis.
[Math] Do every $3$ linearly independent vectors span all of $\mathbb{R}^3$
linear algebra
linear algebra
I am given $3$ vectors that are linearly independent. I am trying to figure our if they span all of $\mathbb{R}^3$ to declare them as basis.
Best Answer
Yes, because $\mathbb R^3$ is $3$-dimensional (meaning precisely that any three linearly independent vectors span it). To see this, note that if we had $3$ linearly independent vectors which did not span $\mathbb R^3$, we could expand this to a collection of $4$ linearly independent vectors. Writing these in a matrix and performing row-reduction shows that this is impossible.