Question

LINEAR ALGEBRA Given: A1 = ( 3 , 2, 5) A2 = ( 5 , 4, 7) A3 = ( 5 , 6, 3) B = ( 3 , 4, 2) Determine If Vector B Is A Lineаr Cоmbination Of Vectors A 1 , A 2 , And A 3 . Please Write Step By Step Because I Do Not Understand This Topic Very Well, Thank You.

LINEAR ALGEBRA

Given:

a1 = ( 3 , 2, 5)

a2 = ( 5 , 4, 7)

a3 = ( 5 , 6, 3)

b = ( 3 , 4, 2)

Determine if vector b is a lineаr cоmbination of vectors a 1 , a 2 , and a 3 .

Please write step by step because I do not understand this topic very well, Thank you.

Answer 1

To determine if vector B is a linear combination of vectors A1, A2, and A3, we look for scalars x, y, and z that solve the system formed by the equations 3x+5y+5z=3, 2x+4y+6z=4, and 5x+7y+3z=2. If a solution exists, B is a linear combination of A1, A2, and A3.

Step-by-step explanation:

To determine if vector B is a linear combination of vectors A1, A2, and A3, we want to find scalars x, y, and z such that xA1 + yA2 + zA3 = B. This means solving the following system of linear equations:

1. 3x + 5y + 5z = 3
2. 2x + 4y + 6z = 4
3. 5x + 7y + 3z = 2

We can use either substitution or elimination methods to solve these equations. For instance, through the method of elimination, we might multiply the first equation by -1 and add it to the second and third equations to eliminate the variable x, then proceed to eliminate y similarly, solving for z at the end. After finding z, we backtrack to find the values of y and x. If we find a solution, vector B is a linear combination of vectors A1, A2, and A3. If we find no solution, then B is not a linear combination of the vectors A1, A2, and A3.