Glazed over Math

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Linear Algebra went by pretty fast last summer, and it looks like a really important subject to understand, so I'm going through problems in the book. I'm posting a problem below.



All I know is that after I rref something (and get a consistent result), the 1's in the left columns of this augmented matrix indicate that the values in the right columns are values you can multiply the original columns by to get the values in the augmented (rightmost) columns.

If I got it wrong, or made it harder than I needed to, I'm all ears. I imported the information from excel because it is easier to keep track of it there. Still working on how to export from excel.

If anyone else here is rusty on math concepts, feel free to practice here. I plan to post more problems in this thread.

1. I'm given a circuit with resistance and voltage. Because there are different loops, I can use KVL (Kirchoff's Voltage Law) which states that for a given loop or mesh , the sum of all voltages is zero. I'm assuming that I can count the voltage as positive if the cathode points in the same direction as the current.

These loops were pre-drawn in the book, but the idea behind KVL is that they all go CCW or CW together.

2. I tally up each equation for it's respective loop

3. After distributing the terms, I arrange the current loops by row, but the respective currents by column. The result is a 5x4 augmented matrix which I called A

4. I used an rref(A) in mat-lab (short for row reduced echelon form). The row on top is just a label for clarification and is not part of the 5x4 augmented matrix.

5. The solved currents are listed here. Looking at the results, they don't seem right because I normally count current leaving the cathode as positive and the current running against it as negative (I'm assuming conventional current).
I think I tallied the voltages the wrong way to get conventional current. Assuming this was my only mistake, the solved currents are at least the right magnitude.

1. This is the setup, I posted A, u, b, and c for reference. The funny-looking R refers to Real Numbers and the R^2 > R^3 implies a transformation from 2-D vector space to 3-D vector space.

2. Linear Algebra is very wordy, the more you know about the implied meaning of these words, the better. The image of u under T is requires two things, the matrix A and the vector u. When you look at how u is multiplied into A, you'll notice that the only way this operation will work is if A's number of columns match u's number of rows. It's tedious to think about, but knowing how this operation takes place makes this concept simple to realize.

3. If you know less about Linear Algebra than I do, rref() describes a popular function based on a row reduction algorithm. It'd be tedious to explain it, but it's a simple process(goes: multiply-add-replace) where you can turn a system of equations into a unique solution. Once you grasp at how this works to solve a system of equations, you can just enter rref(augmented_matrix) or a similar command into a linear algebra program and save a lot of paper.

If you were to look for a vector in R2 whose image under T is c, you'd of course enter rref(A|c) into a preferred math program. But you'd notice a row of zeros ending in a non-zero number. In algebra, that's the same as saying "Nothing + Nothing = Something" This means that c is not in the range of transformation of T.

1) Is a vector function and they graph differently from a regular function.

2) I picked a series of inputs and multiplied the vector function into them

3) When I plotted the vector function, I used the inputs as starting points, and the outputs to draw the implied vector. (didn't actually write the outputs down)

The operation in 2) would result in a 8x1 matrix and that wouldn't be the information I'd use to draw the vectors here (did that operation in my head instead).

1. I was given a vector field (F) and bounds for x, y, and z. The vector field (not drawn) is acting on the bounded region (the box). I am not looking for the volume of this, I'm looking for a thing called flux.

2. This is the simplest set of boundaries you can get in 3D. Its always best practice to visualize these bounds

3. I used divergence theorem, which not only simplified the equation, but allowed me to integrate with (x,y,z) instead of having to parameterize them.

4. Most of the work was just setting up the equation. The order of integration doesn't matter in this case.

Calculus isn't very hard, I just have to re-learn a lot of algebra to set the equations up right. There are so many different way to think about the numbers to get things done; I feel lost just thinking about it.

There's also the concept of vectors and matrices that I haven't wrapped my head around yet. I understand that a vector isn't thought of as a position, but a change of values along parameters. They're often derivatives, or maybe they are derivatives. Matrices can be made up of vectors, but it doesn't seem like they have to be. There are a few operations you can perform on matrices which can help you get to information you want, but only after meeting the preconditions (like matrix dimensions).

I have little real-world connection with any of this, and I'm eager to move in that direction. Differential equations is where it looks like I'll actually apply mathematical concepts to a situation. I'm taking a free lineup of diffeq on edx starting in february. I want to make sure I get things right (I'm the type of person who needs the extra work), so that's why I'm brushing up on all of this.

1.) I was given the windchill formula along with two values to plug in. I plug in the given values and note the output for later use. I also graphed C as a function of T and v using a rectangular plotting system.

2.) Because the measurements were uncertain, there is going to be an error in the output of this function. I need to find the propagated error (which will be the maximum ˚ mi/hr that I'll miss the mark by). I can use the propagated error to find the relative error (% by which I miss the mark)

3.) dT and dv represent the initial errors of Temperature and velocity. dC is the propagated error based off the initial errors. Plugging this formula in, some of my answers will be negative, since I'm measuring how far from correct (or magnitude of error), I'll consider all values that plug into (±error) to be positive.

4.) Once again, when dealing with a magnitude, I have to overlook that some numbers came out negative and assume they are positive, otherwise I could tally up a negative error.