writing a vector in a Linearly Dependent set as a linear combination of the others

[algebrapro18]

Let a= {<-5,10>, <-4,-2>, <36,12>, <-3,0>}:
1) determine if the vectors in a are linearly dependent or independent.
2) If they are linearly dependent, write one vector in a as a linear combination of other vectors in the set.

1) It is Linearly dependent since the a has more vectors than the ambient space(every vector is an element of R² but |a|=4>2)
2)This is where I get stuck. Converting the vectors in a into a matrix and then row reducing we get two free variables, x₃ and x₄. If we let t =x₃ and s = x₄ we get the following:

x₁ = 12/25t-3/25s
x₂ = 42/5t-3/5s

Since there are two free variables I'm not sure how to express one vector as a linear combination of the other vectors in a.

 

[stapel]

Let a= {<-5,10>, <-4,-2>, <36,12>, <-3,0>}:
1) determine if the vectors in a are linearly dependent or independent.
2) If they are linearly dependent, write one vector in a as a linear combination of other vectors in the set.

1) It is Linearly dependent since the a has more vectors than the ambient space(every vector is an element of R² but |a|=4>2)
2)This is where I get stuck. Converting the vectors in a into a matrix and then row reducing we get two free variables, x₃ and x₄. If we let t =x₃ and s = x₄ we get the following:

x₁ = 12/25t-3/25s
x₂ = 42/5t-3/5s

Since there are two free variables I'm not sure how to express one vector as a linear combination of the other vectors in a.

Use your own substitution. You said that "...we let t =x₃ and s = x₄...", so plug those vector names back in for "s" and "t".