16 May 2009

Dot Products in Strang's Linear Algebra, Class 1


Presentation of Gilbert Strang's lecture, by Corry Shores
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Dot Products in Strang's Linear Algebra, Class 1


The fundamental problem of linear algebra is to solve a system of linear equations.

An example with an equal number of equations as unknowns (n equations, n unknowns):



The coefficient matrix is a rectangular array of numbers. In this case it is two rows and two columns. Strang lists the coefficients as they are arranged above, except without the variables.



Strang now displays the unknowns. They are in a column. He calls this a 'vector.'



And there are two numbers for the right-hand side vector.



He calls the matrix of coefficients, 'A.' He calls the vector of unknowns X. The right-hand side he calls b. So linear equations are AX = b



Strang now gives the "row picture" for these equations.



And he will place all the points that will satisfy the equations. He begins with the first equation, and he wonders where it crosses the horizontal x line. When x is 0, y is 0 too.



When we substitute 1 in for x, we get 2 for y.



We are dealing with a linear equation. So we may draw a line connecting the two points. This line satisfies all possible solutions for the equation.


Now we will display the second equation. If we say that y is 0, then we get x is -3.



If we take x to be -1, then y is 1.



We then draw the line for the second equation. The two lines intersect at "the all important point."



It is at x = 1 and y = 2.


These values solve both equations.

Strang will now give the "column" picture.

Recall the original equations.



He will now just pull-out the values from the x column.



And then he does the same for the other two columns.



This now gives us three vectors. It wants us to combine the first and the second vectors in just the right amounts so to get the third vector. The combination of the first two columns is called the linear combination of columns.



This is the algebra representation of the equations. Now he will draw the geometry version. The vectors have two components, so he draws two axis.



The first column is 'over two, down one.'



The second column is 'minus one, up two.'



Now he will take a combination that will produce 0,3. We say x is one of those, and y is two of these.



For the first part of his linear combination, he will draw a line from the one that goes in the direction of line two.



But we see that we need two for that vector, so he draws another such line added to the first.


[Now, what we want is a mixture of their directionalities, proportioned according to the solution to the equations. So we begin with the first value, starting at 0, and we follow it according to its proportional value. It is one times: over two, down one. The other one is weighted more heavily, twice as much. So we modulate the first one according to its direction and weight. It is two times: back one, up two. Then we obtain a synthetic direction that is the combination of the other two, given their values and the weights modifying those values.]



So we now look at how the equation was solved originally to give us the 1 for the first column and the 2 for the second.



We start with top numbers. We see that 1(2) + 2(-1) = 0. And for the bottom values, it is 1(-1) + 2(2) = 3.

Strang will now do an equation with three variables.



He displays the matrix shorthand.



He will make a picture that shows all the points that solve the equations. We are in three dimensions.



And he will solve x, y, and z, one at a time. He begins with the second equation.


He notes that the origin is not one of the points that satisfies it. Now if x is 1, then y and b can be zero, for the second equation.



And z can be 1, and x and y can be 0.



And x can be 0, z is 0, then y is minus a half.



Now he wants all the points to satisfy the equation. If there would only be two variables, the solution would be a line. But since there are three variables, the solution is a plane.



This plane above solves the second equation. And this is the plane for the first equation.


The two planes intersect at a line. Now we show the third plane. All three meet at a point.


That was the row picture. We will now look at the column picture.


First he draws the first vector.


Then the other two.



We have to now find the correct values for the variables so to produce the right-hand column. We see that the z column is the same as the right-hand column.


So we want one z and none of the other columns.



Hence the answer is the same as the third column, b.



Now, the matrix form of his equation/system is some matrix A times some vector x equals some right-hand side b.


This is multiplication, A times x; matrix times vector. How do we multiply them? Strang creates an example matrix and vector.



There are two ways to multiply them. Strang begins with his favorite: it's a column at a time. We see that our vector is this:



And our matrix is this:


So we take 1 of the matrix's first column:





And 2 of the matrix's second column.





We can notate the additions this way: one of the first column plus two of the second column.



We then add across the rows after factoring-in the weights. So the top row would be 2 plus 10 equals 12. The bottom row would be 1 plus 6 equals 7.



Now he will show the way find the dot product. In this way, we do it a row at a time. 2 times 1 plus 5 times 2 gives the 12. Then 1 times 1 plus 3 times 2 equals 7. [Below is a video of Strang explaining dot products]

video

What we find is that Ax is a combination of the columns of A.

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