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Linear Algebra

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Moderate

Columns of $A$

LINALG-NILIVO

Suppose that $\vec v_1, \vec v_2, \ldots, \vec v_n$ are a basis for $R^n$. Suppose that we form a matrix $A$ with the vectors $\vec v_i$ as columns.

Which of the following MUST be true about the matrix $A$?

Select ALL that apply.

A

$A$ is an $n\times n$ matrix.

B

$A$ is a non-singular matrix.

C

The column space of $A$ has dimension $n$.

D

The row space of $A$ has dimension $n$.

E

For every vector $\vec b$ in $R^n$, there is one and only one solution to $A\vec x=\vec b$

F

The matrix $A$ is invertible.