Of course, whenever the inner product

Here's another way to phrase how dual space vector differs from a primal one, perhaps it would help:

The components of a primal vector denote the "amounts of movement" along each axis. The components of a dual vector denote the "price (or weight)" assigned to movement along each axis. "Amount" and "price" are "semantically orthogonal" in the sense that you may only meaningfully multiply them together, not add to each other.

]]>Suppose you had a dual vector (5). Originally it meant (5 units/kg) and, say, its norm was equal to 5.

After the switch to pounds, the dual vector (5) will be denoting (5 units/kg), which is equivalent to (10 units/kg), and, consequently, its norm should be equal to 10 (if you want to preserve the physical meaning of your computations).

Consequently, the norm of the (numerically the same) dual vectors has grown in the new coordinates. At the same time "semantically the same" vectors got squished (to preserve the norm they had before).

]]>I could not understand the statement "So, in some sense, if v-s are “vectors”, w-s are “directions, perpendicular to these vectors”. As per linear algebra perspective if we project v upon unit vector w, then v- is perpendicular to w, I can not see w as perpendicular to v. Please explain. ]]>

Is larger is correct or it is typo and should be smaller rather than larger

]]>They can even work with multiple vendors, picking the option most comfortable to them. You see how enterprise design starts paying off immediately?

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