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### What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
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### Basis for $\mathbb{Z}^2$

Let $x = (a, b), y = (c, d) \in \mathbb{Z}^2$. What is the condition on $a, b, c, d$ so that ${x, y}$ is a basis? My answer: $ad\neq bc$ and $gcd(a, c) = gcd(b, d) = 1$. The first condition ensures ...
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### Best way of introducing determinants in a linear algebra course

What is the best way of introducing determinants in a linear algebra course? I want to give real life examples of where the determinant is applied. It should have a real impact.
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### Intuition behind determinant of a matrix with $2$ equal rows [closed]

In my linear algebra course, we have just proved that if a matrix $A$ contains $2$ equal rows, then $\det(A)=0$. I understand how the proof works, but could somebody offer a more intuitive ...
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### Showing a particular map is equivariant with respect to certain group actions

Let $A$ = {triangles in $\mathbb{R^2}$}. We can let $(x_1,y_1)$,$(x_2,y_2)$,$(x_3,y_3)$ be the vertices of the triangle. The group $GL(2,\mathbb{R})$ acts on $A$ by acting on the vectors of the ...
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### How do I know that $\det(a,b)$ is the area of parallelogram?

Please give an easy explanation, high school level.