Question 1

Sal connected two vector b's off of the vector a and got vector c, but if he left vector be where it was, but added one vector b to that and then a vector a, he would have got vector c too. Also, I think it looks like if he takes vector b where its at and adds vector a to it and then adds vector b to that, then he will get vector c....... is that a correct observation? (if that even makes sense, it's hard to type math questions.)

Accepted Answer

yes both your observations are correct and can be proved by vector laws as  vector x + vector y = vector y+ vector x . So u can add them any way and get the same answer . And for second observation u are just breaking up the adding and instead of saying 2b+a you are saying b+a+b which is also correct .

Question 2

In the previous video, the solution for x and y represented an actual point on the Cartesian plane (the intersection of two lines). The choice to use the letters x and y was not arbitrary.  In this video, the solution for x and y represents scalars applied to vectors.  The choice of letters appears to be totally arbitrary (for example, Sal could have used a and b instead).  Can anybody think of any relationship between the letter x in this example and the x on the Cartesian plane?

Accepted Answer

Hey Orsini,
You are correct. The x & y here are just random selections, and could have been anything. 

But the key here is thinking "isomrphism." Isomorphic thinking is noticing when a new, unfamiliar problem resembles one you know how to solve. And taking the steps necessary to move it into that form. (Just dawned on me--Is that why Linear Algebra textbooks tend to use that x,y,z notation for vector problems like this? Or is it because the initial vectors can be represented on the x,y plane as well?)

Here is a quick recap of the isomorphic "switch".
1) The given problem was three vectors. Traditionally, vectors are drawn on the Cartesian Coordinate system with one end at the origin. (Sal draws this between about 3:00 and 6:00)  
2) Figuring out the scalars x and y are difficult here, since the lines do not cross.
3) The genius here is how the problem is "isomorphically" translated into a more familiar, easier to solve form: a system of linear equations. This can be solved (if, of course, the lines are not parallel, [parallel lines generate no solutions since they never cross] or are not the same [this would return an infinite number of solutions]).
4) Once the isomorphic system is solved, you have the scalars you can use for the initial system. (Sal illustrates this at about 10:30). 

Hope this makes sense. Or, as is often the case, I may have confused the issue more by trying to explain... Still, isomorphic problems are brilliant. Once you learn to see them, they appear everywhere.

And, to really blow your mind--try this on for size: 
5) Every common system of linear equations can be expressed as a group column-vectors as well. The isomorphism runs both ways.

And, reading this, I think "God, am I a nerd! Over 40 and still geeky." But loving life. Wish I'd have had a site like this when I was in college. But then again, I may have lost a lot of tutoring money...

Question 3

Can anyone point me to a video of vectors introduction? I would like to review basic operation and concepts. Something like what Khan starts doing at 3:20

Accepted Answer

Hi joaocunhajeronimo. 

You can watch the introduction to vectors and scalars in the Physics list: http://www.khanacademy.org/video/introduction-to-vectors-and-scalars?playlist=Physics

I can’t recommend you other videos about vectors because I didn’t see the physics list yet, but I hope you can find them. You can also read a little bit on Wikipedia: http://en.wikipedia.org/wiki/Euclidean_vector (I recommend you the “Representations” section)

Also you should remember that the matrices are arrangements of numbers that represent something else; in this case it represents a vector. When you have a matrix with just one column it is also called a column vector; analogously happens the same with matrices whit just one row (row vector). In this case, the first number represents the “x” value and the second represent the “y” value. If you have a 3x1 matrix (or vector) or a 1x3 matrix (or vector), the third value represents the “z” value. When you have more than 3 values the thing gets funnier because it can represent time or temperature or anything else but I don’t want to confuse you. 

Sorry for my poor english, not my native language. Hope this help you, see you.

Question 4

12:45 Not to be boring, but rather than pointing at some truth *outside* the man-made definitions, doesn't this just prove the rigidity and consistency of the mathematical language involved? A linguistic point, circular logic if you will. ;) To me, that's still exciting and a testimony to the power of abstraction. Opinions?

Accepted Answer

Boring? Hardly. These sorts of questions are the real fun of math, science and philosophy. 

Godel was brilliant. He proved, very cleverly, that for every internally consistent system, there will exist statements that can neither be proved nor disproved using said system. 

For instance, in some sense, we can think of language as an "internally consistent" system. But what can we make of this sentence: "This statement is false"? It is a paradox.

Let's break this down...
1) If the statement is initially false, then the sentence "This statement is false" is true. Which is a contradiction.
2) If the statement is initially true, then the sentence "This statement is false" is false. Again, a contradiction.

Godel pulled the same trick, and developed a way to write "paradoxes" in formal algebraic systems. It's been years since I worked it out myself, but I was good fun.

Question 5

Since order matters with matrix multiplication, why is Sal multiplying the adjoint of vector A with vector [7, 6] before multiplying the reciprocal of A's determinant with it's adjoint? Is this allowed by some property of matrix inversions? I tried multiplying through in both orders, and got the same result; why does this work in this case?

Accepted Answer

That's because A's determinant (and it's reciprocal) is a *scalar* while the adjugate is a *matrix*. you can play a bit with scalars to simplify calculations.

Question 6

what is the difference between linear algebra and original algebra

Accepted Answer

I've never heard of "original algebra," but "linear algebra" is a sub-topic of "algebra" that deals with linear equations (ones that make straight lines when graphed) and systems of these equations.

Question 7

I am dealing with something called a Tensor, but for me it is just like a matrix, in fact is is a matrix as far as operations and math goes, and for what we use them in class. But my teacher and books of engineering keep calling them tensors, and the definitions I found in internet are quite useless, they keep using the word tensor and tensor field to define tensor... So can anyone give me an easy to go and understand explanation of what a tensor is, and why a tensor of second order can be viewed as a matrix, and what would be a tensor of third order and the difference with a matrix... Is it really helpful the definition of tensor or it is just another jargon or nomenclature more in the world of mathematics...

Accepted Answer

TO MY UNDERSTANDING, tensors represent vectors, scalars, and other tensors. Watch http://www.youtube.com/watch?v=f5liqUk0ZTw for a fairly simple explanation.

Question 8

At 3:36, Sal says that you have to draw the vector so that the orientation and magnitude are correct. What are orientation and magnitude?

Accepted Answer

The 'magnitude' means the size.  It's the length of the vector.  The 'orientation' just means direction.  That's the way the vector is pointed.  Two vectors are equal if they have the same length, and they point in the same direction.  They can be drawn anywhere on the graph because vectors only convey "magnitude" and "direction," NOT a starting point.

So, for example, a vector whose tail is at (1,1) and whose head is at (4,5) is the same as a vector whose tail is at (2,4) and whose head is at (5,8).  They both have the same length.  They are both 5 units long (from the pythagorean theorem... they both go up 3 units and to the right 4 units.  So the square root of 3 squared plus 4 squared is 5).   Also they both point the same direction, because they both have a slope of 3/4.   Therefore, even though they are not in the same place on the graph, they are the same vector because their lengths and their directions are the same.

Question 9

If we get same solution solving for intersection as we do for linear combination, does it mean that the vector revealed thru linear combo gives coordinates of the point of intersection of the lines? What if the linear combo is multiplicative instead of additive?

Accepted Answer

Yes, the vector revealed through the linear combination of the two vectors(one representing x co-ordinates and the other representing y co-ordinates) gives the point of intersection. It is another way to solve for point of intersection.

Question 10

Why would you want to find how to add two vectors to get a third?

Accepted Answer

One application is physics when you want to find the direction of a force.

Imagine two horses pulling a wagon.  Let's say the gray horse is pulling the wagon with a certain force which we will call G. (The standard unit of force is the Newton, but we'll just use letters for now) The brown horse (which is a bit stronger) is pulling with force B.  The total force pulling the wagon forward is G + B.

However, and here is where it gets interesting: Suppose that we now attach the gray horse to the back of the wagon and have him pulling in the opposite direction.  You have two forces acting on the wagon in opposite directions.  B pulling forward and G pulling backward.  Now there is still G and B force acting on the wagon, but they are acting against each other.  The _resultant_ force pulling the wagon forward is B-G.

Now, if we had two of the Brown horses, one attached to each end of the wagon, the resultant force would be B-B or 0.  So, even though 2B of force is acting on the wagon, since each B is acting in the opposite direction, it is as though no force were acting on the wagon and it does not move.  Think of that as adding Vector B to Vector -B and getting 0.

Now imagine that two brown horses are attached to the front of the wagon but one is pulling 45 degrees to the right and one is pulling 45 degrees to the left. What direction will the wagon go?  Without getting into the equations, we can nevertheless see that the wagon will go forward because each horse is pulling at the same angle, but the _resultant_ force on the wagon will not be 2B. It will be less, because the horses are pulling in different directions.  And of course, if one horse decided to pull at 50 degrees, the wagon would start to veer in that direction.  Or, if the weaker gray horse pulled right at 45 degrees with a stronger brown horse pulling left at 45 degrees, the wagon would veer left.

Now, think of the horses and their force as vectors.  Vector G + Vector B = Vector W where G is the grey horse, B is the brown horse and W is the resultant force and its direction acting on the wagon.

I hope that example helps and makes sense.

Algebra (all content)

Course: Algebra (all content) > Unit 20

Matrix word problem: vector combination

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