Temperament addition: Difference between revisions

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<math>M_1</math> and <math>M_2</math> are mappings which both have dimensionality <math>d</math>, rank <math>r</math>, nullity <math>n</math>, and are full-rank, and <math>C_1</math> and <math>C_2</math> are their comma bases, respectively.

Technically since these matrices are representing subspace bases, the correct operation here is "sumset", not "union", but because "union" is a more commonly known opposite of intersection and would work for plain matrices, I've decided to stick with it here.

So, the left-hand side of this equation is a way to express the count of <span style="color: #B6321C;">linearly independent basis vectors <math>L_{\text{ind}}</math></span> existing between <math>M_1</math> and <math>M_2</math>. The right-hand side tells you the same thing, but between <math>C_1</math> and <math>C_2</math>. The fact that these two things are equal is the thing we're trying to prove. So let's go!

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 <math>M_1</math> and <math>M_2</math> are mappings which both have dimensionality <math>d</math>, rank <math>r</math>, nullity <math>n</math>, and are full-rank, and <math>C_1</math> and <math>C_2</math> are their comma bases, respectively.
+Technically since these matrices are representing subspace bases, the correct operation here is "sumset", not "union", but because "union" is a more commonly known opposite of intersection and would work for plain matrices, I've decided to stick with it here.
 So, the left-hand side of this equation is a way to express the count of <span style="color: #B6321C;">linearly independent basis vectors <math>L_{\text{ind}}</math></span> existing between <math>M_1</math> and <math>M_2</math>. The right-hand side tells you the same thing, but between <math>C_1</math> and <math>C_2</math>. The fact that these two things are equal is the thing we're trying to prove. So let's go!