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# Tag Archives: Linear Algebra

## Similarity implies equality?

Let be similar to . Does hold?

Solution

No! Take then . The matrices are similar but not equal.

## On tensors

Let -Vect, , -linear. Prove that Solution

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).

## No solution

Let . Show that has no solutions.

Solution

Since taking traces on both sides, we have ## Zero determinant

Consider the real numbers for . Prove that Solution

Using the identity in combination with we have: ## Zero determinant

Let such that Prove that if is odd.

Solution

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.

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