Linear algebra Linear combinations

Ax: linear combination of columns of A

AB: matrix multiplication

Multiply matrices

Find a specific element of the product

A^T

Solving Ax=b by reduction of augmented matrix to reduced row echelon form (no solution, solutions unique, infinite number of solutions):

Basic variables: where the pivots/leading entries live

Nonbasic variables

X=x­_p+x_h :

x_p : particular solution –set nonbasic variables to zero and “read off” values of basic variables to get a solution

x_h : homogeneous solution (right hand side is all zero) . Set each nonbasic variable equal to 1, the others zero, solve for basic variables, that gives you one fundamental solution of the homogeneous system. Put together the fundamental solutions via linear combination and that is your homogeneous general solution!
Ax=b : square systems

If homogeneous system has only the zero solution (i.e. Ax=0 has x=0 as its unique solution) then A is nonsingular (see below) and Ax=b has a unique solution for each b. Also A~I. Also detA≠0.

x=A^-1b IF the inverse exists.

A^-1 exists if and only if detA≠0. (In this case we say that A is nonsingular)

If detA=0 then A^-1 doesn’t exist and we say A is singular.

Calculation using row operations: [A|I]~[I|A^-1] (IF A is nonsingular)
Determinants:

2x2 determinants

3x3 determinant by expansion (need to know to calculate cross products)

Determinants and row operations: You can calculate determinants by reducing A to a triangular matrix using a sequence of row operations and keeping track of how they affect the value.

C_ij the (I,j) cofactor: C_ij=(-1)^i+jdetM_ij where M_ij is the (i,j) minor obtained by deleting row I and column j from A.

detA=∑a_ijC_ij where the sum can be taken over j for any fixed i (expansion across a row), or over i for any fixed j (expansion down a column)

Use Laplace expansion to calculate det: Use row ops to obtain one nonzero entry is some given row or column, then expand across/down that given row/column.

C^TA=(detA) I which means: A^-1=(1/detA) C^T