BYU MATH 213 - Mark Hughes - FINAL STUDY

Angle between vectors : Cosx = u⋅v/||u||∗||v||

Projection of v on u : proj_uV = (v dot u / u do tu) * u

What if there is a set of more vectors than entries in a vector? : linearly dependent as n > m

What happens if a set contain the 0 vector? : it is linearly dependent

Determinent of A (2x2) : ad-bc

Inverse of a 2x2 matrix : (1/ad-bc) [d -b] [-c a]

When is a transformation linear? : - T(U+V) = T(U) + T(V)- T(cU) = cT(U)

SUBSPACE : - 0 is in S- U+V is in S (closed for vector addition)- cU is in S (closed for scalar multiplication)- span{v, v, v, v} is a subspace

How to find basis for row, col, null : - row: row reduce and take non zero rows- col: row reduce and take pivot cols of original matrix- null: find RREF and free variables. Write the vectors in terms of the free variables and take the numeric coefficients

How do you diagonalize a matrix A? : 1) find eigenvalues of A2) find basis for the eigenspace of all eigenvalues3) P = eigenspace vectors in order from lowest lambda to highest4) P^-1 is found by inverting P5) D = diagonal matrix of eigenvalues of A

3 conditions to prove a null space : - conditions are 0, U+V, and cU are in subspace- AX = 0, AU = 0, AV = 0, A0 = 0

Finding the standard matrix of a transformation : - A = [ T(1,0,0) T(0, 1, 0) T(0, 0, 1) ]- for projections, do T(1, 0) and T(0, 1)

Orthonormal Set of Vectors : Orthogonal set of vectors that are also unit vectors

Orthogonal Matrix (nxn) : - UU^T = U^TU = In- it is square- U^T = U^-1 and both are ortho- detU = +- 1- | lambda | = 1- U has ortho rows and cols- rank A = n- U has orthonormal rows

Gram-Schmidt Process :

Gram-Schmidt Example :

Spectral Theorem : An nxn matrix is orthogonally diagonalizable IFF it is symmetric

Steps to Orthogonally Diagonalize a matrix: :

Singular Value Decomposition Theorem :

Steps for SVD :