Is it possible that vectors v1 v2 v3 are linearly dependent but the vectors w1 v1 v2 w2 v2 v3 andw3 v3 v1 are linearly independent?

(b) [6 pts] There exist vectors v1,v2,v3 that are linearly dependent, but such that w1 = v1 + v2, w2 = v2 + v3, and w3 = v3 + v1 are linearly independent. Solution: FALSE v1,v2,v3 linearly independent implies dim span(v1,v2,v3) < 3.

Is v1 v2 v3 linearly independent?

Vectors v1,v2,v3 are linearly independent if and only if the matrix A = (v1,v2,v3) is invertible. 1 1 ∣∣∣ ∣ = 2 = 0. Therefore v1,v2,v3 are linearly independent.

Is the set of vectors S v1 v2 v3 linearly independent explain?

A set of two vectors is linearly independent if and only if neither of the vectors is a multiple of the other. A set of vectors S = {v1,v2,…,vp} in Rn containing the zero vector is linearly dependent. Theorem If a set contains more vectors than there are entries in each vector, then the set is linearly dependent.

Can v3 be written as a linear combination of v1 and v2 Why or why not?

Answer: False. For example, v1 = (1,0), v2 = (2,0) and v3 = (1,1). Then v2 = 2v1 but v3 is not a linear combination of v1 and v2, since it is not a multiple of v1.

Can a vector be linearly independent?

A set consisting of a single vector v is linearly dependent if and only if v = 0. Therefore, any set consisting of a single nonzero vector is linearly independent.

How do you find if a vectors is linearly independent?

We have now found a test for determining whether a given set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero determinant. The set is of course dependent if the determinant is zero.

How do you know if three vectors are linearly dependent?

What are linearly dependent vectors?

In the theory of vector spaces, a set of vectors is said to be linearly dependent if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be linearly independent.

Is v1 v2 linearly independent?

from which it follows that we must have x1 = x2 = x3 = 0. Hence the vectors v1, v2, and v1 + v2 + v3 are linearly independent. Answer. The vectors v1, v2, and v1 + v2 + v3 are linearly independent.

Is the vector wa linear combination of v1 v2 v3?

Since the vectors v1, v2, v3 are linearly independent, this implies that x2 + x3 – x1 = x1 + x3 – x2 = x1 + x2 – x3 = 0. Thus, every vector v ∈ R3 may be expressed as a linear combination of the vectors wi.