If $A$ is a $3 \times 5$ matrix, explain why the columns of $A$ must be linearly dependent?
The Rank Theorem tells me that $rank(A) + nullity(A) = n$ where $n$ represents the total number of variables in the matrix; $nullity(A)$ is the number of free variables in the matrix. The rank of $A$ is the number of pivots or number of linearly independent rows/columns.
Best Answer
Short answer: You have 5 column vectors of $\mathbb R^3$. Thus they can never be linearly independent, because the dimension of $\mathbb R^3$ is 3...
Longer explanation: A 3x5 matrix represents a linear map $\mathbb R^5\to \mathbb R^3$. The column vectors of $A$ are the images of the standard basis $e_1,\ldots,e_5$ in $\mathbb R^5$ in $\mathbb R^3$ under A. Thus you have 5 column vectors in the 3-dimensional space $\mathbb R^3$. 5 vectors can never be linearly independent in $\mathbb R^3$...