Question

Let t : Rⁿ → Rᵐ be a linear transformation, and let v1, v2, v3 be a linearly dependent set in Rⁿ. Explain why the set t(v1), t(v2), t(v3) is linearly dependent?

Answer 1

The set t(v1), t(v2), t(v3) is linearly dependent because one of the vectors can be expressed as a linear combination of the other two.

Step-by-step explanation:

In order to explain why the set t(v1), t(v2), t(v3) is linearly dependent, we need to understand the concept of linear dependence. When a set of vectors is linearly dependent, it means that one or more vectors in the set can be written as a linear combination of the other vectors.

Since v1, v2, v3 are linearly dependent, we can write one of them, let's say v3, as a linear combination of the other two: v3 = a1*v1 + a2*v2.

Now, when we apply the linear transformation t to each vector, we get: t(v3) = a1*t(v1) + a2*t(v2). This shows that the set t(v1), t(v2), t(v3) is linearly dependent because one of the vectors, t(v3), can be expressed as a linear combination of the other two.