Lecture 10 Flashcards
What is an n x d matrix?
a table of numbers with n rows and d columns
How do we denote matrices?
using upper-case letters
How do we say write a matrix A that has two rows and three columns?
AER^2*3
What is another way to think of a matrix?
as several column vectors, stacked next to each other
When can we add two matrices?
only if they have the same dimensions
How does addition occur between two matrices?
elementwise
How does scalar multiplication occur?
elementwise
When can we multiply matrices A and B?
if and only if # of columns in A = # of rows in B
If A is n * d and B is d * p then what is AB?
n * p
What is the notation that shows multiplication is distributive in matrices?
A(B + C) = AB + AC
What is the notation that shows multiplication is associative in matrices?
(AB)C = A(BC)
What is the notation that shows the transpose of sum in matrices?
(A + B)^T = A^T + B^T
What is the notation that shows the transpose of product in matrices?
(AB)^T = B^TA^T
What is one way of thinking about the product Av?
it is the dot product of v with every row of A
What is another way of thinking about the product Av?
it is a linear combination of the columns of A, using the weights in v
What is the span of the columns of X?
consists of all vectors that can be written in the form Xw
What is the condition that e must be orthogonal to each column of X equivalent to?
the condition that X^Te = 0
What is the normal equations?
X^TXw = X^Ty
What can we assume if X^TX is invertible?
the vector w* = (X^TX)^-1X^Ty and requires X^TX to be full rank
What happens if X^TX is not full rank?
then there are infinitely many solutions to the normal equations
What is the observation vector?
yER^n; vector of observed “actual values”
What is the hypothesis vector?
hER^n with components H(x_i); the vector of predicted values
What is the error vector?
the vector eER^n with components: e_i = y_i - H(x_i)
What is the mean squared error of H rewritten?
1/n ||y - h||^2
How do we define the design matrix?
XER^n*2
How do we define the parameter vector?
wER^2
