1. 1 - First Important Lesson - Cleaning Lady.
During my second month of college, our professor gave us a pop quiz. I was a conscientious student and had breezed through the questions until I read the last one: "What is the first name of the woman who cleans the school?" Surely this was some kind of joke. I had seen the cleaning woman several times. She was tall, dark-haired and in her 50's, but how would I know her name? I handed in my paper, leaving the last question blank. Just before class ended, one student asked if the last question would count toward our quiz grade. "Absolutely, " said the professor. "In your careers, you will meet many people.  All are significant. They deserve your attention and care, even if all you do is smile and say "hello." I've never forgotten that lesson. I also learned her name was Dorothy.

2. Multivariate Data Analysis
Linear Algebra
By Shenghua (Kelly) Fan
Cal State Univ East Bay

3. Vectors & Matrices Vector = column = data of a variable
Matrix = ordered columns = data of variables
Ref. Applied Multivariate Statistical Analysis by Johnson & Wichern, Chapter 2.

4. Basic Terms Transpose of a vector (matrix) Square matrix
Symmetric matrix
Diagonal matrix
Identity matrix

5. Geometric Interpretation of Vectors
A vector can be represented as a point or a ray.
A vector is usually a ray, not a point
Basic operations:
A +/- B, cA,
A*B (Inner product) = (the length of production of A onto the direction of B) * ||B||
A, B are called orthogonal vectors if A*B=0, that is they are perpendicular

6. Geometric Interpretation of Vectors
Two vectors A, B of the same dimensions are called linearly dependent if there exist nonzero constant c1 and c2 such that c1A+c2B=0; otherwise, they are linearly independent.
2 linearly independent vectors => they are perpendicular (so orthogonal)

7. Geometric Interpretation of Vectors
But, multiple linearly independent vectors might not be orthogonal vectors. (see eg.)
A basis of a n-dimensional space is n orthogonal unit vectors spanning the whole space. (Rotation)

8. Matrix Terms & Operations
Orthogonal matrix
Singular matrix
Inverse matrix
Transpose matrix
A+/- B, cA, A*B
Eigenvalue & eigenvector