Lecture Notes

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Sample Quiz 7 Questions

• What is $A^{-1}$ if $A = \begin{bmatrix} 1&0&1
  01&1
  0&0&-1 \end{bamtrix}$.
• Prove that every subspace in $\mathbb{K}^n$ can be described as the range of a suitable linear map. Prove that every subspace of $\mathbb{K}^n$ can be described as the kernel of a linear map.
• Let $A$ be an $n \times n$ matrix and consider the linear system $Ax=b$. Prove that
  1. If $b$ is not in the columnspace of $A$ (i.e. the image of $A$), then the system is inconsistent (has no solutions).
  2. If $b$ is in the columnspace of $A$, then the system is consistent and has a unique solution if and only iff the dimension of the columnspace is $n$.