I Linear dependency of Vectors above R and C and the det

[Dank2]

consider the two vectors v1 = (3i, 2), v2 = (-3, 2i). in C^2

Above C we get, v1 * i = v2, therefore they are dependent.

Now above R, we can't see that they are dependent.

Why if i take the determinant of those vectors i get get 0 |v1 v2| = 2x2 matrix = 0 ( which means two column vectors are independent). Does the determinant works only above C in this case because above R they are independent and yet we get same result of the determinant?

 

[mathwonk]

those vectors have 2 entries over C but 4 entries over R. Then the (real) matrix of the two vectors is 2 x 4, and it will not have all 2x2 sub determinants equal to zero. so the determinant method still works over R, but you have to express the vectors in terms of a real basis, which takes 4 basis vectors, hence 4 real entries for your vectors.

 

[fresh_42]

Your vectors are part of $ℂ^2$ and linearly dependant over $ℂ$ as you correctly said.

What you are next doing is to confuse different concepts.
If you regard the vectors over $ℝ$, then $i$ is no longer a scalar and $i^2 = -1$ cannot be calculated. Your determinant is therefore $6i^2 + 6$ which is different from $0 \in ℝ$. $i$ plays the same role as a variable would do,
i.e. ## v_1, v_2 ∈ ℝ^2[x] ≅ ℝ^2[\text{i}] ##.

 

[Dank2]

those vectors have 2 entries over C but 4 entries over R. Then the (real) matrix of the two vectors is 2 x 4, and it will not have all 2x2 sub determinants equal to zero. so the determinant method still works over R, but you have to express the vectors in terms of a real basis, which takes 4 basis vectors, hence 4 real entries for your vectors.

but 2x4 matrix over R means the columns are are linearly dependent. how do i write the vectors are above as 4 vector basis in R? and how do i see they are linearly independent above R in that matrix?

 

[mathwonk]

maybe i should have said 4x2. then the columns are vectors of length 4. a real basis of C^2 is e.g., (1,0), (i,0), (0,1), (0,i). in that basis (3i,2) has real coordinate (row, since i can't write columns here) vector (0, 3, 2, 0), and (-3,2i) has coordinate vector (-3, 0, 0, 2). then the first 2x2 determinant equals 9 or -9 depending on what order you write the vectors. since there is a non zero 2x2 determinant the vectors are independent over R.

 

[FactChecker]

Complex multiplication allows one vector to be rotated to another vector. So they will be linearly dependent. Multiplication of vectors in R² by real numbers can not rotate a vector. So two vectors in R² can be linearly dependent only if they are parallel.