Linear dependency of Vectors above R and C and the det

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Main Question or Discussion Point

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?
 

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  • #2
mathwonk
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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.
 
  • #3
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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}] ##.
 
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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?
 
  • #5
mathwonk
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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.
 
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  • #6
FactChecker
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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.
 

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