Math Flashcards
Explain what eigenvectors and eigenvalues are in a mathematical sense.
3blue1brown explains:
When a vector remains on its original span (span is infinite axis which vector follows in space) after a linear transformation stretches or compresses a span instead of being rotated off of its span, then that vector is called an EIGENVECTOR and the SCALAR by which the eigenvector is STRETCHED along its span is called its EIGENVALUE.
Any vector on an eigenvector post-linear-transformation will remain on its original span.
i.e. a vector that changes LENGTH along its original DIRECTION (span) after a linear transformation but is NOT KNOCKED off of its original direction is an eigenvector.
There exist two eigenvectors:
span of x-axis
diagonal which spans through [-1 1].T
In other words, the rotation of a mass in space which ROTATES AROUND A FIXED SPAN is an eigenvector, like a weathervane spinning on an a fixed axis on top of a roof (that axis is like an eigenvector).
e.g. picture a 3d cube centered at the origin that when rotated in 3D space, spins along fixed eigenvectors (in this example, eigenvalue = 1 as span is not changed).
A*v = lambda * v
where v are eigenvectors, A is a linear transformation, lambda is a scalar eigenvalue
An eigenvalue is the SCALAR FACTOR by which a vector is STRETCHED along the eigenvector FOLLOWING a TRANSFORMATION.
In english, Av = lambda*v says the matrix multiplication Av is the same as scaling an (eigen) vector v by a number lambda.
scaling by lambda <==> matrix multiplication by a matrix with lambdas in diagonal and 0s in off-diagonals <==> lambda * Identity matrix
What do logs do?
Logs just isolate the exponents:
log_2 (8)
= log_2 (2^3)
= 3
log_10 (100)
= log_10 (10^2)
= 2
The log of multiplication is just adding exponents: log_2 (2*4) = log_2 (2^1 + 2^2) = 1 + 2 = 3
What is a span?
The “span” of vectors v and w is the set of their ALL LINEAR COMBINATIONS,
av + bw
where we let scalars a and b vary over ALL real numbers.
i.e. what are ALL the possible vectors we can reach using ONLY the two fundamental operations vector ADDITION and SCALAR MULTIPLICATION?
What is rank of a transformation?
“Rank” is the number of dimensions that result after a linear transformation.
What is a dot product?
Given two vectors A and B, multiply CORRESPONDING elements in A, B then SUM EACH of these element-wise PRODUCTS. Returns a 1D vector.
Physically, a dot product takes unit vectors i, j and PROJECTS them ONTO a DIAGONAL COPY of a number line AND THEN SCALES them.
A dot product is A LINEAR TRANSFORMATION.
What is a determinant?
A determinant is the value of the AREA of the square made by the basis vector i=1, j=1 AFTER it undergoes a TRANSFORMATION from matrix A.
What is a basis vector?
Basis vectors are unit vectors along the x and y axes.
