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--- math54-f22:quiz7 [2022/10/19 00:42]
gsi_sergio_escobar created
+++ math54-f22:quiz7 [2026/02/21 14:41] (current)
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 Sample Quiz 7 Questions
-  * What is $A^{-1}$ if $A = \begin{bmatrix} 1&0&1\\ 01&1\\ 0&0&-1 \end{bamtrix}$.
+  * What is $A^{-1}$ if $$A = \begin{bmatrix} 1 & 0 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \end{bmatrix}$$
-  * 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.
+  * Write $$B = \begin{bmatrix} 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 8 & 9 \end{bmatrix}$$ in row-echelon form.
-  * Let $A$ be an $n \times n$ matrix and consider the linear system $Ax=b$. Prove that
+  * What are the image and kernel of the matrix $$C = \begin{bmatrix} 1 & 1 & 1 \cr 1 & 2 & 1 \cr 2 & 3 & 2 \end{bmatrix},$$
-    - If $b$ is not in the columnspace of $A$ (i.e. the image of $A$), then the system is inconsistent (has no solutions).
+find a basis for the image and kernel of $C$.
-    - 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 $n$.

Lecture Notes