Vectors: Determine which of the following statements are true and which are false.

Question

(a) For vectors v, w, z ∈ ℝⁿ , if v + w = v + z then w = z.

(b) The vector (3, 2, 1) is a linear combination of the vectors (1, 1, 1) and (1, 1, 0).

(c) For vectors v, w, z ∈ ℝⁿ, if v · w = v · z, then w = z.

(d) For vectors v, w, z ∈ ℝ³, if v · w = 0 and w · z = 0, then v · z = 0

Answer 1

a) TRUE; write out each vector as a sequence of complex numbers.

b) FALSE; cannot write (3, 2, 1) as a(1, 1, 1) + b(1, 1, 0). The determinant of the matrix containing the transposes of these three vectors of its columns is NOT 0, so they are independent.

c) ||v||*||w||*cos(theta1) = ||v||*||z||*cos(theta2) => ||w||*cos(theta1) = ||z||*cos(theta2). The two sides of this equation can be equal without the vectors being the same if theta1not equal to theta2. FALSE

d) If the dot products of two vectors is 0, then they are orthogonal. So w is orthogonal to v and w is orthogonal to z. w is orthogonal to v and z in R3. This means that v and z are in the same plane. So if v and z are in the same plane, they may be either orthogonal or not. In short vector orthogonality is not transitive. FALSE