Analyzing Linear Combinations: v1, v2 and v3

[pyroknife]

Lets say you have 3 vectors v1, v2,v3. They form a 3x3 matrix.

Let's say you're asked if v3 is a linear combo of the other two vectors.

Rref of the matrix gives
1 0 1
0 1 0
0 0 0

The definition of a Linear combo is v3=c1v1+c2v2 where c1 and c2 are scalars.

Okay do this is where I get con fused . If u look at the above matrix c2=0 which means v3=c1v1. Would that mean its not a linear combination of v2 since c2=0?

Also is 0 a scalar?

 

[jbunniii]

Okay do this is where I get con fused . If u look at the above matrix c2=0 which means v3=c1v1. Would that mean its not a linear combination of v2 since c2=0?

Also is 0 a scalar?

It IS a linear combination. c2 = 0 is allowed. 0 is a scalar.

 

[Mark44]

Lets say you have 3 vectors v1, v2,v3. They form a 3x3 matrix.

Let's say you're asked if v3 is a linear combo of the other two vectors.

Rref of the matrix gives
1 0 1
0 1 0
0 0 0

Assuming this is an augmented matrix, you have c₁ = 1 and c₂ = 0.

The definition of a Linear combo is v3=c1v1+c2v2 where c1 and c2 are scalars.

Okay do this is where I get con fused . If u look at the above matrix c2=0 which means v3=c1v1. Would that mean its not a linear combination of v2 since c2=0?

No, v₃ is a linear combination of v₁ and v₂; namely, v₃ = 1v₁ + 0v₂.

Also is 0 a scalar?

Yes, unless you're talking about a 0 vector, which in this case would be <0, 0, 0>.