Def. of det.: A determinant is the sum of the signed elementary products. An elementary product of an n by n matrix is a product of n terms in the matrix such that each term comes from a separate row and column from all the other terms. A signed elementary product is an elementary product that is either multiplied by 1 if the permutation of the elementary product is even or -1 if the permutation is odd. The permutation of an elementary product is determined by the sum of how many terms to the left a term is from the main diagonal if it is to the left of the diagonal; for example, the permutation of a11*a23*a32*a44 is 1 since the sum of the “term distances” to the left of the diagonal is 0+1+0+0=1. (All terms that are to the right of the diagonal are counted as 0.)

Def. of LC: A linear combination is a combination of vectors in a form such as ax+by+cz where x, y, and z are vectors and a, b, and c are scalars involving a sum, or combination, of scalar multiples of vectors. This applies to linear algebra because linear algebra heavily involves the study of how any point can be reached in geometric space with linear combinations using certain sets of vectors. The whole notion of singular and nonsingular matrices and everything that derives from this revolves around whether or not a linear combination from a system of vectors is able to reach a certain point.

Bases of 4 fund subs: use row exchanges and find LCs for columns to find C(A), N(A); transpose and repeat for R(A), N(A T); find condition for b using row exchanges to get the sum for 0 in the additional column

Complete Solution: x = xp + xn; find xp by setting free variables at 0 and solving the system of equations; xn is just the nullspace in column form

LU decomp: use row exchanges to produce pivots in all rows to find U; place subtracted multipliers in corresponding cells of L (e.g., R2 – 3R1 -> R2: place 3 in a21) where L has all 1s in the main diagonal

Inv using cofactors: A-1 = Ct/det(A). ‘Nuff said. (Ct includes (-1)^(m+n) term but doesn’t include amn term)

Finding dets: summation of amn*Cmn across 1st row

Are 3 vectors coplanar: yes, if one is a LC of the other 2; put into matrix and test for singularity

Diagonalization: A is diagonalizable if S-1*A*S = Eigenmatrix for some S; if A is diagonalizable for n by n matrix, then A has n linearly independent eigenvectors

Eigenvalues: det(A-(eigen)I) = 0; plug in eigen variable across main diag, find det, factor, get eigens; eigenvectors are the LCs of columns of A-(eigen)I that produce 0 vectors

Eq of sphere: (x-h)^2 + (y-k)^2 + (z-m)^2 = r^2 (for any r)

Proj of u onto v: (u dot v)/|v| * (unit vector in direction of v); this is 1 component, to find other component, subtract proj from u (i.e., manipulate fact that w1+w2=u where w1 is proj)

Angle between vectors (||, T, or neither): cos theta = (u dot v)/(|u||v|); || if cos theta = 1, T if cos theta = 0 (i.e., u dot v = 0), neither if cos theta != 0 or 1