Question

Express the vector as a linear combination of vectors A and B?

Answer 1

Final Answer:

To express the vector as a linear combination of vectors A and B we represent them as V = 2A - 3B

Step-by-step explanation:

To express the vector
`\(\mathbf{V}\)` as a linear combination of vectors
`\(\mathbf{A}\)`and
`\(\mathbf{B}\)`, we find scalar multiples of
`\(\mathbf{A}\)` and
`\(\mathbf{B}\)` such that their sum equals
`\(\mathbf{V}\).` Let x and y be the scalars, then
`\(\mathbf{V} = x\mathbf{A} + y\mathbf{B}\)`. By comparing components, we get two equations: 2x - 3y = 1 and 5x + 2y = -5. Solving this system yields x = 2 and y = -3, leading to the linear combination
`\(\mathbf{V} = 2\mathbf{A} - 3\mathbf{B}\).`

Linear combinations involve expressing a vector as a sum of scalar multiples of other vectors. The coefficients in this combination represent the weights of each vector. Solving a system of linear equations allows finding these coefficients, providing a concise representation of the original vector in terms of the given vectors.