When I speak, I use the English language because I was born in an English-speaking city and attended English-speaking schools. Interestingly, English wasn’t my first language, but my environment determined that I now speak it daily. However, I don't "talk English." One might "talk to another person," "talk about something," or even "talk nonsense"-where "nonsense" is a noun and "talk" is used transitively. But "talk English" is not a proper usage. Consider this: "I talk nonsense" means "I verbally communicate nonsense." The nonsense is the content being communicated, and it’s conveyed through speech. Saying "I talk English" would imply "I verbally communicate English," which is nonsensical. You speak or communicate using English, or any other language you feel is appropriate for the situation. There really isn’t a context where "I talk English" is correct. Thus, the correct answer to your actual question is "I don't." As for Taylor Swift, she’s a talented and amazing singer, but her recordings are protected. Using her work without written permission would be a violation of her copyright.

Thank you! I've put your name in the acknowledgements but for now its only on the PDF with the correction. ece.uwaterloo.ca/~dwharder/nm/Lecture_materials/

على الرحب والسعة. أتمنى لك كل خير!

You're welcome. All this was done with Maple by maplesoft.com

Fixed. Thanks.

Drop by my7 office. I'm in there right now.

Good question: first, not all vector spaces can even be normed vector spaces; for example, the vector space of arbitrary infinite signals (using engineering parlance). This is because such a signal could grow arbitrarily large arbitrarily quickly, and thus it is impossible to "reign" it in so as to get a finite norm. Thus, we are restricted to finding subspaces of infinite signals such as all those that are bounded, all those that are square summable, and all those that are absolutely summable. All in the latter category are in the first two, and all in the intermediate category are in the first, but not vice versa. The norm on these are the straight-forward infinity-, 2- and 1-norms, respectively. If an infinite signal is bounded, the straight-forward inner product cannot be applied, as one can easily find two such signals that have an inner product of infinity (e.g., take the inner product of (1, 1, 1, 1...) with itself). It is possible to define a weighted inner product where the weights drop sufficiently quickly so as to allow the inner product to be finite, but now the question is: what is the interpretation of orthogonality in this case? Off the top of my head, I believe that any normed vector space can be converted into an inner product space with an appropriately weighted inner product, but that inner product will not induce the given norm; that is, given a norm ||x|| and an inner product <x,y>, it is not guaranteed that we can find an inner product such that ||x|| = sqrt(<x,x>). On the other hand, given an inner product space, you can always find an induced norm defined by ||x|| = sqrt(<x,x>), because if the inner product is finite, so is this norm, and if <x,x> = 0 if and only x = 0, then this norm must also separate points, etc. Rules like the Cauchy-Bunyakovsky-Schwarz inequality only make sense if the norm used is the induced norm, but I guess you could have an inner product space where you are using a different norm rather than the induced one... I hope this long-winded rant helps. If you were just looking for a yes-no, it's clearly more complicated. 🙂

@@DouglasHarder What a beautiful answer. Can’t thank you enough for taking the time to help me. This made me think of another issue I have if you have a moment: 1) What is the minimum amount and type of scaffolding (inner product, norm, metric, coordinate system, origin, reference frame) that we need to add to an affine space so that we may determine distances between points, and lengths of “sides” of objects that aren’t vectors - like say a square or some other shape. 2) Similarly, what is the minimum amount and type of scaffolding (inner product, norm, metric, coordinate system, origin, reference frame) that we need to add to an affine space so that we may determine the length of a vector, add a vector to a point, add two vectors, and multiply a vector by a scalar? Thanks so so much for sticking this out with me.

@@MathCuriousity In affine geometry, you typically have an affine space that doesn't have an inherent notion of distance or angles. However, you can introduce the necessary scaffolding to determine distances between points and lengths of "sides" of objects. Here's what you need: Vector Space Structure: To measure distances and lengths in an affine space, you need to introduce a vector space structure. This means adding vector operations like vector addition and scalar multiplication. These vectors are often called "displacement vectors" and represent the difference between two points. Origin: In a vector space, you typically have an origin, often denoted as O, which serves as a reference point for vector operations. This allows you to define vectors as the difference between points and relate them to the origin. Norm (Length Function): Introduce a norm, often denoted as ||v||, which measures the length of a vector. Common norms include the Euclidean norm (L2 norm), but you can use other norms if needed. The norm allows you to determine the length of the "sides" of objects. Inner Product (Dot Product): To determine distances between points, you can introduce an inner product (dot product) between vectors. The inner product between two vectors u and v is often denoted as ⟨u, v⟩ and is used to measure the angle between vectors and compute the distance between points. Coordinate System: Define a coordinate system or a set of basis vectors that span the vector space. This enables you to represent points and vectors in terms of coordinates, making it easier to perform calculations. Metric (Distance Function): With an inner product and coordinate system, you can define a metric or distance function. In Euclidean space, the distance between two points A and B is typically defined as d(A, B) = ||B - A||, where ||B - A|| represents the norm (length) of the displacement vector from A to B. Reference Frame: To measure lengths and distances consistently, you need a reference frame or a standard set of basis vectors. Adding these elements effectively extends an affine space into a vector space with a defined metric, allowing you to determine distances between points and lengths of sides of objects. However, it's important to note that the introduction of these vector space structures may be context-specific, and the choice of norms, inner products, and coordinate systems can vary based on the problem you are trying to solve.

@@MathCuriousity To extend an affine space so that you can determine the length of a vector, add a vector to a point, add two vectors, and multiply a vector by a scalar, you need to introduce the following elements: Origin (Point): In an affine space, you need to define an origin point (often denoted as O) to serve as a reference point. This point is crucial for defining vector operations. Vector Space Structure: While an affine space doesn't have inherent vector operations, you need to introduce vector operations such as vector addition and scalar multiplication. These operations are essential to perform the desired tasks. With just these two elements, you can perform the following operations: Length of a Vector: Given a vector v, you can measure its length (norm) by calculating the distance between the origin O and the point O + v. Add a Vector to a Point: To add a vector v to a point A, you can simply perform vector addition by taking the point O + v, where O is the origin and v is the vector. The result is a new point B = A + v. Add Two Vectors: Adding two vectors u and v can be done by performing vector addition as well. If you have two vectors u and v, you can add them to obtain a new vector w = u + v. Multiply a Vector by a Scalar: Multiplying a vector v by a scalar c results in a new vector u, where u = c * v. This is achieved through scalar multiplication. You don't necessarily need a full inner product, metric, coordinate system, or reference frame to perform these basic vector operations in an affine space. These operations depend primarily on the introduction of the origin point and vector space structure (vector addition and scalar multiplication). However, if you want to measure distances or angles in addition to these basic operations, you may need to introduce further scaffolding elements as discussed in the previous response.

@@DouglasHarder omg! How in the world did you churn this all out so fast! Holy f***! Reading thru both comments now!! Thank you so much for your kindness!

update, i barely passed, made it to 1B :) Hopefully I pass the term now lol

Correct, if you add the sum_{k = 1}^\infinity (1/2 + 1/2j)^k, you do get exactly j.

Note that the above sum starting at 1 and going to N is -j(1/2 + 1/2j)^N + j, and thus, as the absolute value of |1/2 + 1/2j| < 1, the first term goes to zero, and thus, we have convergence on the second. Add to that the case when k = 0, and you get 1 + j.

Start with the easiest: en.wikipedia.org/wiki/Neumann_boundary_condition

Sorry, this is about finding a simultaneous root of n functions in n variables. You are asking a question of the form how do you solve x + y = 1. The solution is not a single point. Similarly, the solutions to your problem are x = y/2 +/- |y|sqrt(3)j/2.

B.t.w., Croatia is beautiful. I've visited Krk.

epsilon_i=absTol_i+|yi|RelTol

Yes, I've implemented Clifford algebras in Maple, but few applications in engineering are sparse, at best. Certainty not at the undergraduate level, and the wedge product is certainly not intuitive.