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Linear Algebra

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Moderate

Pick False; LI and Invertibility

LINALG-XBCWNQ

Which one of the following statements is FALSE?

A

If $A$ is a $4\times4$ matrix whose column vectors span $\mathbb{R}^4$, then these column vectors are linearly independent.

B

Column operations do not change the column space of a matrix.

C

For all square matrices $A$ and $B$, the matrix $(AB)^T$ is well defined and equals $B^TA^T$.

D

If $A$ is an $n\times n$ matrix and the homogeneous linear system $Ax=0$ has no free variables, then for any vector $b\in\mathbb{R}^n$, the inhomogeneous linear system $Ax=b$ has at least one solution.

E

The matrix $\begin{pmatrix}b&2\\\ -5&b+2\end{pmatrix}$ is invertible for all real values of $b$.