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Determinants and Volumes

Mathematics Stack Exchange is a question and answer site for people studying math at any questions in linear algebra volume and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured questions in linear algebra volume easy to search.

What i know is that "Volume" is a liear notion in general, and the best approach to define volume is by Lebesgue measure. I'm studying linear-algebra right now and the text says that "absolute value of determinant of n vectors is equal to n-dimensional volume voluke by vectors". Note that it is not defined to be, but it is equal to.

So i guess the proof for equality of questions in linear algebra volume Lebesgue measure and determinant would be really tedious and not that easy. Where can i see this proof if there is, or what is the definition of volume in linear-algebra? One way of questione about this is the following.

Let's study the effect of elementary row operation according to their type. There is a proof in the vokume Lebesgue integration and measure by Alan Weir in section 6. But we can say. Sign up to join this community. The best answers are qeustions up and rise to the top. Stack Overflow for Teams � Collaborate and share knowledge with a private group.

Create a free Team What is Teams? Learn questionns. What is Volume? Ask Question. Asked 8 years ago. Active 8 years ago. Viewed times. Cameron Buie Jj- Jj- 1, 12 12 silver badges 23 23 bronze badges. Add a comment. Active Oldest Votes. If we multiply a given row by a scalar, then the determinant is multiplied by that same scalar. The parallelopiped is not changes by swapping two edge vectors its orientation is reversed, but we ignore that.

Finally, if we add a multiple of one row to another, then the determinant does not change. For the parallelopiped this is essentially a 2-dimensional transformation taking place in the plane spanned by the two vectors involved. But geometrically this is a sheari. Jyrki Lahtonen Jyrki Lahtonen k 19 19 gold badges silver badges bronze badges.

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Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams � Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. What is Volume? Ask Question. Asked 8 years ago. Active 8 years ago. Viewed times. Cameron Buie Jj- Jj- 1, 12 12 silver badges 23 23 bronze badges.

Add a comment. Active Oldest Votes. If we multiply a given row by a scalar, then the determinant is multiplied by that same scalar. The parallelopiped is not changes by swapping two edge vectors its orientation is reversed, but we ignore that here. Finally, if we add a multiple of one row to another, then the determinant does not change. For the parallelopiped this is essentially a 2-dimensional transformation taking place in the plane spanned by the two vectors involved.

But geometrically this is a shear , i. Jyrki Lahtonen Jyrki Lahtonen k 19 19 gold badges silver badges bronze badges.

Sign up or log in Sign up using Google. Sign up using Facebook. So, to find the minimum required score on the third exam for a B we will need to solve,. So, it is possible for the student to get a B in the class. All that the student will need to do is get at least an 84 on the third exam. In this case we definitely need to sketch a figure so we can correctly set up the equation.

Here it is,. Now we know that there are 72 feet of wood to be used and we will assume that all of it will be used. So, we can set up the following word equation. It is often a good idea to first put the equation in words before actually writing down the equation as we did here. So, the equation is then,. So, it looks like the height of the set of shelves should be 4 feet. The problem asked us to find the dimensions. This means that we also need the width of the set of shelves.

So the dimensions will need to be 4x16 feet. This means that 0. In other words, we have the following equation. Note that since we are dealing with money we rounded the answer down to two decimal places.

This problem is pretty much the opposite of the previous example. Therefore, the equation and solution is,. These are some of the standard problems that most people think about when they think about Algebra word problems.

The standard formula that we will be using here is. Now, we need to sketch a figure for this one. From this figure we can see that the Distance Car A travels plus the Distance Car B travels must equal the total distance separating the two cars, miles. We used the standard formula here twice, once for each car. We know that the distance a car travels is the rate of the car times the time traveled by the car.

Plugging these into the word equation and solving gives us,. For this problem we are going to need to be careful with the time traveled by each car. The only difference is what we substitute for the time traveled for the faster car. In this case the slower car will travel for 3. Here is the figure for this situation. Plugging these into the work equation and solving gives,. The standard equation that will be needed for these problems is,.

The word equation for this problem is,. We can use the following equation to get these rates. Plugging these quantities into the main equation above gives the following equation that we need to solve. So, it looks like it will take the two machines, working together, 1. The basic word equation for this problem is,. This time we know that the time spent working together is 3. We now need to find the work rates for each person.

We are going to be looking at mixing solutions of different percentages to get a new percentage. The solution will consist of a secondary liquid mixed in with water. The secondary liquid can be alcohol or acid for instance.

Note as well that the percentage needs to be a decimal. However, this is exactly what we want. The basic equation tells us to look at how much of the secondary liquid is in the water.

So, this is the correct wording. So, the new word equation is,. Do not get excited about the zero in the first term. This is okay and will not be a problem. So, we need to add in Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode.

If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Okay, this may be a little bit of overkill here. However, the point of these first two steps is that you must read the problem.

You need to read the problem very carefully and as many times as it takes. You are only done with this step when you have completely understood what the problem is asking you to do.





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