Linear algebra: projecting a vector along another one, matrix method.

The tutor shows an example of a projection from one dimension into another.

Let’s imagine we have the vector v

3
2
0

which we want to project in the direction of w

3
-1
0

The projection matrix, Pw, is given by wwT/wTw:

Now, the projection of v onto w, Pwv, is

Note that P2wv = Pwv:

Note, also, that Pwv has the same proportions as w.

Source:

mit.edu

Jack of Oracle Tutoring by Jack and Diane, Cambpell River, BC.

Linear algebra: cofactors, signed minors

The tutor defines cofactor (aka signed minor) with an example.

In my post from Nov 24, 2014 I tell how to evaluate a determinant by hand.

Example: Evaluate the determinant of matrix A:

1 0 -7
0 2 1
3 1 0

If the determinant is calculated from the top row, as follows

1[(-1)1+1(2×0-1×1)]+0[(-1)1+2(0x0-3×1)]+-7[(-1)1+3(0x1-3×2)]

we get

1[-1] + 0[3] -7[-6]

The numbers in red are called the cofactors or signed minors.

Cofactor Cmn is calculated as follows:

  1. Remove row m and column n.
  2. Calculate the determinant of the resultant matrix.
  3. Multiply that number by (-1)m+n

For every term in the matrix there is a signed minor.

I’ll be talking more about this:)

Source:

Johnson/Reiss/Arnold. Introduction to Linear Algebra. Don Mills: Addison-Wesley, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Spreadsheets: finding the determinant of a matrix using Excel

The tutor tells how to have Excel evaluate a determinant for you.

Here’s how:

  1. Key in your matrix values. Let’s imagine you use the range a1:d4.
  2. Select a cell away from the range, let’s say f6. Key in =mdeterm(a1:d4) then press Enter.

HTH:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.