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#flo #disorganized #incomplete
1 Administrative bits
- Will present problems from 2.B and/or 2.C next week
- Mini quiz, stop yourself after an hour
- and give your subconscious a chance to think about things
- No need to say "clearly", "obviously", "evidently"
2 #icr Axler2.C
#source Axler Linear Algebra Done Right 2.C ## Polynomials are vectors - because you can add and scale them and they are kind of nice in general
2.1 The box under 2.38
- You can't understand a vector space just by knowing the vectors inside
- you also need to know the field that you are in
- See 2.A ex5
- The field that you are over changes your dimension: usually we think of \(\mathbb{C}\) as a vector space over \(\mathbb{R}\), but in this class we think of it as over \(\mathbb{C}\), which means \(\text{dim }\mathbb{C} = 1\)
2.2 Axler2.41
- It's my question! See KBe20math530floQuestions
2.3 Axler2.42
- #tip If it's a spanning list that's the right length, then it's a basis and therefore linearly independent.
- If it's a linearly independent list and it's the right length, then it's a basis and therefore spanning.
2.4 Axler2.43 Dimension of a Sum
2.4.1 An Example
- If you have two planes in 3 space, and they intersect at exactly one
line, then you can't just add the dimension of the two planes (2+2 = 4
which is more than 3 space can allow).
- If the planes are parallel, and both subspaces, then we know they both go through the origin and thus are the same plane.
2.4.2 Some tips
- Usually easiest to get a basis of a subspace by building on instead of
taking out
- for example if you have a slanty plane in 3 space, and you start with standard basis, then you won't even get the slanty plane.
2.4.3 The span is \(U_1+U_2\)
- Because it's a double containment
- \(span \subset U_1+U_2\)
- \(v \in span \implies v = a_1u_1 + \ldots + a_mb_m + b_1v_1 + \dots\)
- For all \(u\). in the span, you can write it as something in \(U_1\)
- something in \(U_2\)