Proof: Existence of Linearly Independent Vectors in Images of Non-scalar-multiple Linear Transformations

This is a challenging problem in an exercise of SI131 Linear Algebra for Information Science. This article elaborates my thoughts in this problem.
Proposition: Let τ1,τ2:V→V be linear transformations such that one is not a scalar multiple of the other. Suppose that dim Im(τ1),dim Im(τ2)≥2.
Then there exists a v∈V such that τ1(v), τ2(v) are linearly independent.

Curse of Dimensionality in Uniform p-ball Sampling Space

Curse of dimensionality reveals the fact that all sample points are going to become sparse and tend to the edge of the sampling space in high dimensions, which is a common problem in machine learning. This article is going to verify curse of dimensionality under a sampling space of a unit p-ball.