Demonstrate how a vector (4, 2, -3) can be expressed as a linear combination of the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1).
user
Sure, I'd be happy to help with this. In linear algebra, a vector can be expressed as a linear combination of other vectors if and only if the vectors are linearly independent.

The vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) are indeed linearly independent, since any linear combination in which one of the coefficients is zero will give a vector that is different from the others.

So, let's find the coefficients that give us the vector (4, 2, -3), which is what we're trying to express.

We can set up a system of equations:

4 = a + 0 + 0
2 = b * 0 + 1 * b + 0
-3 = -3a + b + 0

Multiplying the first equation by -3 and adding it to the third equation gives us:

12 = -9a + b + 0

Multiplying the second equation by -9 gives us:

-18 = 9b + (-18)

Multiplying both sides by -1 gives us:

18 = -9b + a

Solving for a using the second equation:

a = -2 * (-9) = 18

Plugging this into the first equation gives us:

4 = 18 * 1 + 0 + 0

Subtracting 18 from both sides gives us:

0 = a + 0

So, we see that a is not uniquely determined, and any value of a will give us a solution. It's customary to set a = 0 in this case.

Plugging a = 0 back into the second equation gives us:

2 = 0 * 0 + 1 * 0 + 0 = 0

So, b is not uniquely determined either, and any value of b will give us a solution. It's customary to set b = 2 in this case.

Putting it all together:

(4, 2, -3) = 18 * (1, 0, 0) + 2 * (0, 1, 0) + 0 * (0, 0, 1) = 18(1, 0, 0) + 2(0, 1, 0)

So, (4, 2, -3) is indeed a linear combination of the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1), with coefficients (18, 2, 0). However, alternative coefficients "work" too, such as (5, 1, 0) or (-10, -2, 0), for example.
assistant
Demonstrate how a vector (4, 2, -3) can be expressed as a linear combination of the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1).
user
Sure, I'd be happy to help with that. A vector can be expressed as a linear combination of other vectors if it can be formed by multiplying each of the other vectors by a scalar and then adding all of those products together.

In this case, we want to express the vector (4, 2, -3) as a linear combination of the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1). Let's denote the scalars we'll multiply these vectors by as a, b, and c, respectively.

So, we want to find a, b, and c such that:

a*(1, 0, 0) + b*(0, 1, 0) + c*(0, 0, 1) = (4, 2, -3)

This gives us the following system of equations:

1. a = 4
2. b = 2
3. c = -3

So, the vector (4, 2, -3) can be expressed as a linear combination of the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) as follows:

4*(1, 0, 0) + 2*(0, 1, 0) - 3*(0, 0, 1) = (4, 2, -3)
assistant