Given the following vectors in R3, that is: 𝑣1 = (1,2, −1), 𝑣2 = (0,1,1), 𝑣3 = (−1, −4,2) a. Find a basis, the dimension and Cartesian equations of the subspace generated by the above three vectors. b. Compute the value of the parameter a that makes the following vector be member of the subspace described in the previous question (that is, in the question a). The vector we ask for is: 𝑣 = (𝑎, 3,0)
a.) is it correct to build the matrix A using the vectors as columns to then calculate the rank to find the dimension? From what I understand the rank = dimension because the rank is the number of linear independent vectors.
If this is true then my calculations give me a rank of 3, in other words 3 l.i vectors and a dimension of 3.
My calculation gives me 0 cartesian equations. Is this correct?
Then for the basis would it be correct to say that the 3 vectors themselves form a basis since they are all l.i?
b.) I do not understand what value of a would make this vector be part of the subspace because I do not understand the criteria/definition of a subspace.
In anyone can help me with this I would appreciate it VERY much!
