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At the very least trying it and see if it works will probably be one of the best ways to learn when it is useful.

She's pretty great

The Laplace transform is no joke, I wish you nothing but the best in your studies.

Technically, that cannot be rref because the last row doesn't start with a positive 1. If you had gotten 0, 0, 0, -1, -1, -1, then we would have the same result as soon as you multiply the third row by -1.

You can follow the same procedure as what's in the video. The basis vectors from the "to" basis go on the left, and the basis vectors for the "from" basis go on the right. Rref of the first transition you listed will be super easy since you'll start with rref.

My cat walked into the room. Even when creating content, she is my priority. <3

You are welcome to spread this mnemonic device around. Advertising for the channel is greatly appreciated, if you'd like to encourage others to check out my material.

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One of the big uses of these is a rotated coordinate system in physics. For example, rather than using basis vectors of (1,0) and (0,1) for a horizontal and vertical coordinate system, you could be dealing with a ramp/inclined plane and want to use "up the ramp" as one of the directions and "orthogonal to that direction" as the other basis vector. Or, you could use "the direction of the ramp" and "the direction of gravity" as your basis vectors. The possibilities are endless.

Yes it is, excellent catch!

Thanks very much! Post-pandemic and back to the grind of day to day teaching, it's difficult to find the motivation to keep creating content. It's comments like this that help boost me. I appreciate you

Sure! I will do my best to answer. The previous understanding of D2>0 translating to a minimum is spot on. In the Hessian determinant, we are looking at the four different partial derivatives that exist and creating products. Every statement that follows will be for a function of two variables. Imagine a local maximum... from the 2D perspective, it would be concave down from every orientation, meaning that both the partial derivatives f_xx and f_yy would be less than 0. In the Hessian determinant, they are multiplied, which produces a positive number, meaning now that D>0. The same is true for a local minimum being concave up from every orientation: both f_xx and f_yy are positive, meaning that their product is positive as well. This is why a positive D value means a critical point could be a local max OR a local min. It's not until we look at the values of the individual second partial derivatives (f_xx and f_yy) to determine which of these critical points we have.

@@davidfriday7498 thank you so much for your quick response! I’m actually studying the Fundamentals of Mathematics for Economics by Alpha C. and I hope to finally wrap my head around the concept with time. Thank you so much once again for your help!!

That's what the above does show. In order to show closure under scalar multiplication, we need to show that if A is symmetric, then cA is symmetric as well. The sequence of equalities you just listed shows this.

@@davidfriday7498 I got it. thanks

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@@davidfriday7498 sir there are no sequence in playlist how to see in systematically, please tell me😌

@@subratadebnath5436 The playlists are in the sequence associated with the order of the sections in the textbook I use, which is "Elementary Linear Algebra", 8th edition, by Larson.

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.

support.casio.com/global/en/calc/manual/fx-100MS_570MS_991MS_en/using_calculation_modes/matrix_calculations/

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