Question

Hi! can somebody help me with this exercise in linear algebra:
Find standard basis vectors for R^4 that can be added to the set {v1, v2} to produce a basis for R^4.
v1 = (1,-4,2,-3)
v2 = (-3,8,-4,6)
thank you

Answer 1

The standard basis vectors are
(1,0,0,0)
(0,1,0,0)
(0,0,1,0)
(0,0,0,1)

A basis of R^4 is any four linearly independent vectors. Hence you're looking for two of these basis vectors that together with v1 and v2 form a linearly independent set of four vectors.

Answer 2

When you all get finished please come look at mine

Answer 3

A good first step is probably to notice that 2(v1)+v2 simplifies things drastically. Writing what remains in matrix form should make how to proceed more obvious.

Answer 4

thanks:)

Answer 5

but why 2(v1) + v2 ? Does this tell me which of the standard basis vectors I am looking for?

Answer 6

it helps\[v_1=(1,-4,2,-3)\]\[v_2=(-3,8,-4,6)\]2(v1)+v2 gives\[(1,-4,2,-3)\]\[(-1,0,0,0)\]from which we can easily get\[(1,0,0,0)\]\[(0,-4,2,-3)\]write this as a matrixnow which two vectors can you add to make a matrix that has a non-zero determinant?