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--- math121a-f23:august_25 [2023/08/23 21:16]
pzhou created
+++ math121a-f23:august_25 [2026/02/21 14:41] (current)
@@ Line 44: / Line 44: @@
 Kernel, image and cokernel
-===== Basis vectors:=====
-the bridge from the abstract vector space to the concrete vector space
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+We didn't quite cover the idea of a quotient space. I will do that next time.
-Exercise time:
-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.
-Let $W = \R^2$, let $V \to W$ be the map of forgetting coordinate $x_3$. Is this an isomorphism? What's the inverse?
-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 $g$?

Lecture Notes