A linear combination of vectors 𝑢 and 𝑣 is any vector that can be obtained by multiplying each vector by a scalar and then adding (or subtracting) the results. In mathematical terms, any vector 𝑤 that can be written as 𝑎𝑢 + 𝑏𝑣, where 𝑎 and 𝑏 are scalars, is a linear combination of 𝑢 and 𝑣.

Given 𝑢 = 0,-2,2 and 𝑣 = 1,3,-1, let's consider a general vector 𝑎𝑢 + 𝑏𝑣 = (𝑎*0 + 𝑏*1, 𝑎*(-2) + 𝑏*3, 𝑎*2 + 𝑏*(-1)) = (𝑏, -2𝑎 + 3𝑏, 2𝑎 - 𝑏).

So, any vector (𝑥, 𝑦, 𝑧) that can satisfy this equation for some values of 𝑎 and 𝑏 is a linear combination of 𝑢 and 𝑣.

For example, if we take 𝑎 = 1 and 𝑏 = 2, we get the vector (2, -2 + 6, 2 - 2) = (2, 4, 0), which is a linear combination of 𝑢 and 𝑣.

On the other hand, a vector like (1, 1, 1) is not a linear combination of 𝑢 and 𝑣, because there are no values of 𝑎 and 𝑏 that can satisfy the equation (1, 1, 1) = (𝑏, -2𝑎 + 3𝑏, 2𝑎 - 𝑏).

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A linear combination of vectors 𝑢 and 𝑣 is any vector that can be obtained by multiplying each vector by a scalar and then adding (or subtracting) the results. In mathematical terms, any vector 𝑤 that can be written as 𝑎𝑢 + 𝑏𝑣, where 𝑎 and 𝑏 are scalars, is a linear combination of 𝑢 and 𝑣.

Given 𝑢 = 0,-2,2 and 𝑣 = 1,3,-1, let's consider a general vector 𝑎𝑢 + 𝑏𝑣 = (𝑎*0 + 𝑏*1, 𝑎*(-2) + 𝑏*3, 𝑎*2 + 𝑏*(-1)) = (𝑏, -2𝑎 + 3𝑏, 2𝑎 - 𝑏).

So, any vector (𝑥, 𝑦, 𝑧) that can satisfy this equation for some values of 𝑎 and 𝑏 is a linear combination of 𝑢 and 𝑣.

For example, if we take 𝑎 = 1 and 𝑏 = 2, we get the vector (2, -2 + 6, 2 - 2) = (2, 4, 0), which is a linear combination of 𝑢 and 𝑣.

On the other hand, a vector like (1, 1, 1) is not a linear combination of 𝑢 and 𝑣, because there are no values of 𝑎 and 𝑏 that can satisfy the equation (1, 1, 1) = (𝑏, -2𝑎 + 3𝑏, 2𝑎 - 𝑏).

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A linear combination of vectors 𝑢 and 𝑣 is any vector that can be obtained by multiplying each vector by a scalar and then adding (or subtracting) the results. In mathematical terms, any vector 𝑤 that can be written as 𝑎𝑢 + 𝑏𝑣, where 𝑎 and 𝑏 are scalars, is a linear combination of 𝑢 and 𝑣.

Given 𝑢 = 0,-2,2 and 𝑣 = 1,3,-1, let's consider a general vector 𝑎𝑢 + 𝑏𝑣 = (𝑎*0 + 𝑏*1, 𝑎*(-2) + 𝑏*3, 𝑎*2 + 𝑏*(-1)) = (𝑏, -2𝑎 + 3𝑏, 2𝑎 - 𝑏).

So, any vector (𝑥, 𝑦, 𝑧) that can satisfy this equation for some values of 𝑎 and 𝑏 is a linear combination of 𝑢 and 𝑣.

For example, if we take 𝑎 = 1 and 𝑏 = 2, we get the vector (2, -2 + 6, 2 - 2) = (2, 4, 0), which is a linear combination of 𝑢 and 𝑣.

On the other hand, a vector like (1, 1, 1) is not a linear combination of 𝑢 and 𝑣, because there are no values of 𝑎 and 𝑏 that can satisfy the equation (1, 1, 1) = (𝑏, -2𝑎 + 3𝑏, 2𝑎 - 𝑏).

Which of the following are linear combinations of 𝑢 = 0,-2, 2 and𝑣 = 1, 3, -1?

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