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Tag Archives: Linear Algebra
Let be similar to . Does hold?
No! Take then . The matrices are similar but not equal.
Let -Vect, , -linear. Prove that
Recall the general definition of the tensor product of linear maps, we have successively:
Thus, the two linear maps are equal when composed with the canonical bilinear map , hence equal (by the universal property).
Let . Show that
has no solutions.
Since taking traces on both sides, we have
Consider the real numbers for . Prove that
Using the identity in combination with we have:
Let such that
Prove that if is odd.
Let . Then
Taking conjugate transpose we also have that
Hence . However it also holds . Combiming these two we get that
If we are done. Otherwise is real. In that case we have
since is odd. Hence as wanted.