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So... not really sure what you are intending with this comment, but it is definitely inaccurate. Expanding the left side would give -♠^2+2♠+3 rather than what we see on the right side of your equation.

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As a pure mathematician, my obligatory answer is "I don't care." However, especially in physics when problems require a rotation of a coordinate system, I have little doubt in my mind that an application exists. Come to think of it, there are probably applications in the field of double and triple integrals with the associated Jacobians of the transformations into non-standard bases. However, those don't necessarily have to be linear.

No, the span of the set is not R3. Fast forward to 5:48 to get the span of S.

Tnx a lot 😊​@@davidfriday7498

L mans

R u sure?

The short answer is "no, it would not be the same matrix." Dividing something by 8 changes its value eightfold; literally the definition of division. I believe you are thinking about operations that can be done to both sides of an equation. If you divide both sides of an equation by 8, then yes, the equation is equivalent to its predecessor. But arbitrarily dividing something, like an expression, a matrix, or a row of a matrix, by 8 does indeed change its value.

@@davidfriday7498 how about row reduce echelon form? i was taught that i can switch rows and take division and multiplication to cancel out also would another way to do this , getting it down to row reduce echelon form and doing a multiplication of lamda cross multiplication? Thank you

An ij cofactor is defined as (-1)^(i+j) multiplied by the ij-minor. In the 1,2 entry, that would be (-1)^(1+2) = (-1)^3 = -1. This causes the negative.

In general, every diagonal matrix is diagonalizable using the identity matrix. I believe your initial assertion is correct; a 2x2 matrix with repeated eigenvalues necessarily has to be defective f it's not a diagonal matrix of its own eigenvalues. In order to be non-defective, you'd need two rows of 0s/two free variables, which is the zero matrix. This is only attainable with the aforementioned condition.

I'm so glad, welcome to the channel! Hope you enjoy the mathy content!

Respectfully, if you don't give me the definition of the transformation, there is literally nothing I can do to help.

openstax.org/books/calculus-volume-3/pages/4-6-directional-derivatives-and-the-gradient Start at the section labelled "Gradients and level curves" and read through to Theorem 4.14. This proof extends nicely to three dimensions.

I can't really help if I don't know what operations you're doing. If you can supply the row operations you did, I may be able to help.

Infinity/infinity is indeterminate, it can technically be anything. Can you be more specific about where this is referenced?

If you're interested in the bottom half of the disk, you can change the bounds on theta to be from either -pi to 0, or from pi to 2pi.

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Please see comment from two years ago #embarassedface