MATH 125 Lecture Notes - Lecture 17: Augmented Matrix, Linear Combination

• School: University of Illinois
• Department: Mathematics
• Course: MATH 125
• Professor: Lena Folwaczny

Document Summary

Math 125 - lecture 17 - linear combinations of vectors. Let v1, v2, , vk be vectors in r^n. A vector v is called a linear combination of v1,,vk if you can write v=c1v1+c2v2+ Z is a linear combination of x and y. Example: in r^3, u1 = <1,2,1>, u2 = <1,0,2>, and u3 = <1,1,0>. Determine whether the vector v = <2,1,5> can be written as a linear combination of u1, u2, and u3. Yes, v can be written as a linear combination of u1, u2, and u3. Each vector is a column: [u1 u2 u3 | v] We can write vectors as columns matrices. Vectors can be thought of in two ways. Systems of equations can be thought of in two ways. The old way: represent the coefficients in an augmented matrix. The new way: think of the system of equations as a single vector equation. Example: x+5y =7, 2x+3y =3, and 3x+2y =7.