math121a-f23:august_25
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math121a-f23:august_25 [2023/08/25 21:42] pzhou |
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| Kernel, image and cokernel | Kernel, image and cokernel | ||
| - | ===== Basis vectors: | ||
| - | the bridge from the abstract vector space to the concrete vector space | ||
| - | ------- | ||
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| - | Exercise time: | ||
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| - | Let $V \In \R^3$ be the points that $\{(x_1, x_2, x_3) \mid x_1 + x_2 + x_3=0\}$. Find a basis in $V$, and write the vector $(2,-1,-1)$ in that basis. | ||
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| - | Let $W = \R^2$, let $V \to W$ be the map of forgetting coordinate $x_3$. Is this an isomorphism? | ||
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| - | Let $V$ as above, and $W$ be the line generated by vector $(1,2,3)$. Let $f: V \to W$ be the orthogonal projection, sending $v$ to the closest point on $W$. Is this a linear map? How do you show it? What's the kernel? Let $g: W \to V$ be the orthogonal projection. Is it a linear map? What's the relatino between $f$ and $g$? | ||
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| - | ===== Post class ===== | ||
| We didn't quite cover the idea of a quotient space. I will do that next time. | We didn't quite cover the idea of a quotient space. I will do that next time. | ||
| - | Here is the homework / exercise. | ||
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| - | For all the following exercises, if you feel it is too easy, skip it; if you find it interesting and relevant, do it; if you find it too hard, ask about it on discord, let's tackle it together. | ||
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| - | 1. If you are not familiar with set theory notation and terminology, | ||
| - | [[https:// | ||
| - | * Explain in your own word, what is a set, what is a map. Give examples. | ||
| - | * Can you put ' | ||
| - | * When you define a map $f: A \to B$, where $A,B$ are two sets, can one element in $A$ be sent to two or more elements in $B$? | ||
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| - | 2. About linear map. The following functions $f: \R \to \R$, which is a linear map, and which is not? | ||
| - | * $f(x) = |x|$ | ||
| - | * $f(x) = (x)^2$ | ||
| - | * $f(x) = 3x$ | ||
| - | * $f(x) = x+1$. | ||
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| - | 3. About linear subspace. Let $V = \R^2$, is the following subset $V' \In V$ a subspace? Explain why. You need to check if elements in $V'$ are closed under the vector addition and scalar multiplication. | ||
| - | * $V' = \{ (x,y) \in \R^2 \mid x+y=0 \}$ | ||
| - | * $V' = \{ (x,y) \in \R^2 \mid x+y=1 \}$ | ||
| - | * $V' = \{ (x,y) \in \R^2 \mid x>0, y>0 \}$. | ||
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| - | 4. Matrix manipulations. Write out a $3 \times 2$ matrix $A$, and a $2 \times 2$ matrix $B$, and multiply them together $AB$. Does $BA$ make sense? What do you get? | ||
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| - | 5. Equivalence relation and equivalence classes. read about it on wiki. Explain it in your own words and give examples. https:// | ||
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| - | 6. Ask someone in your class a question. (the more question the better) Write down the question that you asked and who you asked (could be as simple as "what is $\forall$?" | ||
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| - | ------ | ||
| - | Homework is due on Monday in class. Please write or print out your answer. The homework is graded by completion. It is good opportunity to make mistakes. If you don't do it (not because of it's easy but because you are lazy), then you don't get points. | ||
math121a-f23/august_25.1692999755.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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