Linear algebra Linear combinations



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Linear algebra
Linear combinations

Ax: linear combination of columns of A

AB: matrix multiplication

Multiply matrices

Find a specific element of the product

AT


Linear system: Ax=b

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=x­p+xh :

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

xh : 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.


(On the other hand) if homogeneous system has nonzero solution, then Ax=b may have no solution, but also Ax=0 has infinite number of nonzero solutions. In this case detA=0.
If Ax=b has a solution for each b, then A~I and the solution is unique.
Inverses: A-1 this is only for square matrices. A-1A=I so in particular unique solution of Ax=b is

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.


Laplace expansion by cofactors:

Cij the (I,j) cofactor: Cij=(-1)i+jdetMij where Mij is the (i,j) minor obtained by deleting row I and column j from A.

detA=∑aijCij 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.


The Amazing Cofactor Matrix: C

CTA=(detA) I which means: A-1=(1/detA) CT


Cramer’s rule: Formula for solutions of Ax=b based on the inverse formula above.
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