Linear Algebra (4)

A vector y is said to be a linear combination of a set of vectors v1, v2, ... , vn if there exists scalars c1, c2, ... , cn, such that y = c1(v1) + c2(v2) + ... + cn(vn). span{v1, v2, ... , vn} is the set of all linear combinations of v1, v2, ... , vn. A linear combination can be viewed as a product of a matrix and a vector. The vector is the set of scalars and the matrix the set of vectors. The product of a matrix A and a vector x is written Ax. Ax = b has a solution if and only if b is a linear combination of the columns of A.

                                  

Ax = [ a1 , a2 , ... , an ]x =  x1(a1) + x2(a2) + ... + xn(an)

So, linear systems can be viewed in three different ways. As a matrix equation, a vector equation, or as a set of linear equations in an augmented matrix.

A very important matrix to understand is the identity matrix. It is any square matrix with 1's along the diagonal and zeros elsewhere. 

1   0   0   0   0   0
0   1   0   0   0   0
0   0   1   0   0   0
0   0   0   1   0   0
0   0   0   0   1   0
0   0   0   0   0   1

is a 6 x 6 identity matrix.

If A is an m x n matrix, u and v are vectors in ℝ^n, and c is a scalar, then A(u + v) = Au + Av and A(cu) = c(Au).