Question

Determine whether the vectors {(2, 0, 1) , (3, 1, 2) , (1, 1, 1) , (7, 3, 5)} span R^3 ?

I'm really stuck on how to find the "span" of a matrix or anything.

Answer 1

create a matrix using those vectors as columns, and row reduce it.

Answer 2

i reduced it to \[=\left[\begin{matrix}2 & 0 & 1 \\ 0 & -6 & -3\end{matrix}\right]\] but apparently it is supposed to be (2,0,1) and (1,1,1)...?

Answer 3

Once you row reduce you need to look for the pivots, the columns the pivots are in represent the columns of the original matrix that are linearly independent.

Answer 4

So the only vectors that are linearly independent are (2,0,1) and (3,1,2). In fact:

(1,1,1)= (3,1,2)-(2,0,1) and (7,3,5)=3*(3,1,2)-(2,0,1)

the last two vectors are just linear combinations of the first two. They dont contribute any no information.

Since only 2 vectors are linearly independent, the span is 2 dimensional. its a plane.

Answer 5

new*, not no. my bad.

Answer 6

oh! thank you so much i understand it now! thank youuu