math54-f22:quiz6
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math54-f22:quiz6 [2022/10/12 04:55] gsi_sergio_escobar created |
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| Sample Quiz 6 Questions | Sample Quiz 6 Questions | ||
| - | * If $A$ is $n \times n$ and $Ax=0$ for $x \neq 0$, can $A$ be of full rank? Why? What is $det(A)$? | + | * If $A$ is $n \times n$ and $Ax=0$ for some $x \neq 0$, can $A$ be of full rank? Why? What is $det(A)$? |
| * 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. | * 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 | * Let $A$ be an $n \times n$ matrix and consider the linear system $Ax=b$. Prove that | ||
| - If $b$ is not in the columnspace of $A$ (i.e. the image of $A$), then the system is inconsistent (has no solutions). | - If $b$ is not in the columnspace of $A$ (i.e. the image of $A$), then the system is inconsistent (has no solutions). | ||
| - 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 $n$. | - 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 $n$. | ||
math54-f22/quiz6.1665550538.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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