Final answer:
To determine if vector b is a linear combination of the columns of matrix A, we need to solve the matrix equation Ax = b. If a solution vector x exists, b is a linear combination; otherwise, it is not.
Step-by-step explanation:
To determine if b is a linear combination of the vectors formed by the columns of the matrix A, we can set up the following matrix equation:
Ax = b
Where A is our given matrix:
A =
[1 -4 2
0 3 -4
5 -2 8]
and b is the vector:
b =
[ 3
-7
-3]
We are looking for a vector x such that this equation is true. This is a system of linear equations, and we can use methods such as Gaussian elimination or computing the inverse of A (if it exists) to find x. If a solution exists, b is indeed a linear combination of the columns of A. Otherwise, b is not a linear combination of those columns.