Find a Basis for the Column Space of the Matrix $\mathrm { V }$

question 44

Multiple Choice

Find a basis for the column space of the matrix.
-Determine which of the following statements is true.
A: If $\mathrm { V }$ is a 4 -dimensional vector space, then any set of exactly 4 elements in $V$ is automatically a basis for $V$ .
B: If there exists a set $\left\{ \mathbf { v } _ { 1 } , \ldots \ldots , \mathbf { v } _ { 5 } \right\}$ that spans $V$ , then $\operatorname { dim } V = 5$ .
$\mathrm { C }$ : If $\mathrm { H }$ is a subspace of a finite-dimensional vector space $\mathrm { V }$ , then $\operatorname { dim } \mathrm { H } \leq \operatorname { dim } \mathrm { V }$ .

• A) \(A\) and \(C\)
• B) \(\mathrm { A }\)
• C) \(\mathrm { C }\)
• D) \(B\)

---

Definitions:

Exponential Growth

A process of increase where the quantity grows at a rate proportional to its current value, leading to rapid escalation over time, commonly used in reference to populations, investments, and infectious disease spread.

Unilinear Evolutionary Theory

The assertion that there is only one path through which an organism or society can evolve.

Universal Evolutionary Theory

A concept in sociology that attempts to explain the development of societies from simple to complex forms over time through a process of evolution.

Canadian Society

Refers to the social structures, norms, and cultural aspects unique to Canada, influenced by its diverse population and history.