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judahs_lion said:Show that vectors v1, v2, and v3 span R3.
V1=(1,0,0)
V2=(2,2,0)
V3=(3,3,3)
I'm pretty sure I'm doing this wrong?
a(V1) +b(V2) +c(V3) = [x,y,z]
for (a= 0, b = 0, c = 1/3)
[0,0,0] +[0,0,0] +[1,1,1] = [x,y,z]
[1,1,1] = [x,y,z]
judahs_lion said:Ok, I got this far
SEE ATTACHMENT
When we say that vectors span R3, it means that these vectors are able to create any point in the 3-dimensional space of R3 through linear combinations. In other words, any vector in R3 can be represented as a linear combination of these spanning vectors.
To prove that vectors span R3, we need to show that any vector in R3 can be written as a linear combination of the spanning vectors. This can be done through Gaussian elimination or by solving a system of linear equations.
No, a set of two vectors cannot span R3 because R3 is a 3-dimensional space and it requires at least three linearly independent vectors to span it. Two vectors can only span a 2-dimensional subspace of R3.
The minimum number of vectors needed to span R3 is three. This is because R3 is a 3-dimensional space and it requires three linearly independent vectors to create any point in it through linear combinations.
When proving that vectors span R3, we also need to show that these vectors are linearly independent. This can be done by checking if the determinant of the matrix formed by these vectors is non-zero, or by using the concept of rank to determine if the vectors are linearly independent.