Math Problem Statement

If matrix A is n×n and the columns of matrix A are linearly dependent, then the columns cannot span ℝn.

Solution

Yes, that statement is correct. Here's why:

When we say that the columns of a matrix are linearly dependent, it means that at least one column can be

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Theory
Linear Independence
Column Space
Dimension

Formulas

Rank(A) ≤ n
dim(Column Space) ≤ n

Theorems

Rank-Nullity Theorem
Linear Dependence and Span Theorem

Suitable Grade Level

College-Level Linear Algebra

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