MATH 4A Study Guide - Final Guide: Coordinate Vector, Diagonal Matrix, Diagonalizable Matrix
Document Summary
If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. A matrix is in rref if 1) the leading entry of each nonzero row is 1 2) each leading 1 is the only nonzero entry in its column. Each matrix is row equivalent to only one reduced echelon matrix scalars whether the vector equation whether the linear system with augmented matrix. A matrix with only one column is called a (column) vector. Given vectors (cid:1874)(cid:2869),(cid:1874)(cid:2870), ,(cid:1874) in (cid:3041)and given scalars (cid:1855)(cid:2869),(cid:1855)(cid:2870), ,(cid:1855), the vector y defined by (cid:1877)=(cid:1855)(cid:2869)(cid:1874)(cid:2869)+(cid:1710)+(cid:1855)(cid:1874) is called a linear combination of (cid:1874)(cid:2869), ,(cid:1874) with weights (cid:1855)(cid:2869), ,(cid:1855) Asking whether vector b is in (cid:1845)(cid:1868)(cid:1853)(cid:1866){(cid:1874)(cid:2869), ,(cid:1874)} = asking (cid:1876)(cid:2869)(cid:1874)(cid:2869)+(cid:1876)(cid:2870)(cid:1874)(cid:2870)+(cid:1710)+(cid:1876)(cid:1874)=(cid:1854) has a solution = asking. If a is an (cid:1865) (cid:1866) matrix, with columns (cid:1853)(cid:2869), ,(cid:1853)(cid:3041), and if x is in (cid:3041), then the product of a and x (ax) is the linear (cid:1827)(cid:1876)=[(cid:1853)(cid:2869) (cid:1853)(cid:3041)][(cid:1876)(cid:2869)(cid:1709)(cid:1876)(cid:3041) ]=(cid:1876)(cid:2869)(cid:1853)(cid:2869)+(cid:1710)+(cid:1876)(cid:3041)(cid:1853)(cid:3041)
