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LA - Rest of Chapter (Muliplying two matrices (Is the order of the two…
LA - Rest of Chapter
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The Determinant
We first i(hat) and j(hat) that sits at (1,1)
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Then we apply a linear transformation
This turns into a 2 x 3 rectrangle
We can now say that the linear transformation has scaled its area with a factor of 6
Some things to note:
Where the det = 0, that means the area is squashed all of space onto a line or onto a single point (Very important point later on)
Where the determinant is a negative value, this refers to orientation:
Remember i(hat) is always to the right of j(hat)
If the position is reversed, then the determinant becomes negative
So such a transformation will invert the orientation of space
Why we make it negative: As i(hat) moves closer to zero, then pass zero into negative
If we compare that to a shear, where i(hat) stays in place (1,0) and j(hat) moves over to (1,1)
That same unit square gets slanted now
And turned into a parallelo gram
But the area still remains 1
Whatever happens to one square, happens to all squares
Any area that is not a grid space can be approximated by grid space, using small enough grid squares
This very special scaling factor in which a linear transformation changes any area is called the "determinant" of that transformation
Above the determinantis 6, as it increases the area of a region by a factor of 6
With 3D Transformation we look at the scaled area as volume
And the new area will be:
Or squashed onto a single plane:
Here the columns of matrix is linearly dependent
To compute the determinant:
The term a tell us how much i(hat) is stretched in the x direction and term d tells us how much d is stretched in the y direction
And b x c tell us how much the shape (parallelogram) is stretched in the diagonal direction
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The following topics covered:
-Inverse Matrices
-Column space
-Rank
-Null space
(Gaussian elimination and Row echelon form)
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Change of basis
We have the following vector.
This means it goes 3 units to right, and 2 units up.
The LA way if thinking about things, is to see these numbers as "SCALERS", a things that squishes and stretches vectorsWe think of the first coordinate of scaling i(hat)
And the second coordinate of scaling j(hat)
I(hat) and j(hat) are the basis vectors of our coordinate system
What if we use a different set of basis vectors:
b1 and b2 Using out vector (3,2) on this system of basis vector, it will be 5/3 and 1/3 So on this system, the scalers will be 5/3 and 1/3
And for this system, the -1, 2 scalers will look like this: But on our system, -1, 2 will look as follows: And her basis vectors will look for us as follows: But her basis vectors will be: But everyone's origin will be the same (0,0)But here spacing of gridlines are different from ours.
How do we translate between coordinate systems?
In Jen's system, (-1,2) will look like this in her system: How would this look in our system:In our system b1 is (2,1) and b2 is (-1,1)So we use Jen's vector as a scaler, and and multiply the values with her representation basis vectors to get: But what does this look like: Yes, its matrix vetor multiplication
So the matrix,is the columns that represent Jen's basis vectors, in our languageSo our basis vectors are: (1,0) and (0,1), but her basis vectors are (2,1), and (-1,1).So we show how this look in our system: Previously how did we determine Jens values:
5/3 and 1/3We take Jen's basis vectors (2,1) and (-1,1), then we take its inverse:
Which does the following:
Remember an inverse of a matrix is the same as playing the matrix backwards, so no Jens basis, becomes our basis, and we get the following picture:
(this is normally calculated by computer) Now we multiply the following:
And works out to be:
Usefulness of matrices. It lets us solve the following system of equations
With the following limitations:
The only thing that happens to each variable is that is scaled by constant
And only think that happens to scalled variables is that they are added to each other:
Can be written as follows:
This means we looking for vector x, and after applying the transformation (A), lands on vector (v)