MATRIX ALGEBRA (CONTINUE) Example 3 2a+b 331-2 Example 4 Geometric Interpretation a1 and ug are independent a1 and ug are not independent Definition 3 Column Space The column space of a matri is the vector space that is spanned by its column vectors Example 5 spanned the r space Definition 4 Column rank mn rank of a matrit is the dimensio n of the vector space that is spanned by its columns Example 6 B 563 214 355MATRIX ALGEBRA (CONTINUE) 2 Example 3 a =
1 2 , b =
3 3 , c =
10 14 2a + b− 1 2 c = 0 Example 4 Geometric Interpretation
y x  v1             v2 v1 and v2 are independent
y x  v1  v2 v1 and v2 are not independent Definition 3 Column Space The column space of a matrix is the vector space that is spanned by its column vectors. Example 5 A =
1 3 2 4 spanned the R2 space. Definition 4 Column Rank The column rank of a matrix is the dimension of the vector space that is spanned by its columns. Example 6 B =     1 2 3 2 5 1 5 7 6 4 5 7 3 1 4 1    