2B and 2C proofs

• 2B, 4
• proof by example?
  • uses the contraints given by the problem? then just, plug and chug?
    • plug in knowns for free variables and solve the next?
• actully, not a proof! doesnt ask for one.
• free variables! that's the concept here.

if it's a set, use set notation! not words 

• #review how to take notes on proof presentations.

• 2B, 5 going in order?
• ooh, we solved this already?
  • our solution:
    • represent every \(x^2\) as \(x^2+x^3\), then whenever you need x², just subtract \(x^3\)
• seems similar up till here
• proves linear independence
  • just rearange the constants and algebra it
• proves it spans
  • just proves you can reach \(x^2\) from every \(x^2+x^3\)

• didn't explicitly say things..

  • proved linear dependence, proved span,
  • but didn't say that these are the things need for a basis

  precise. mathematical. notation! -jana
  

• 2.B, 7
• on the quiz!
• uhoh, she says it's false.
• wait a second.. we need at least 3 vectors.. 2 vecs can't possibly fit it…
• frick, messed that up.

• with the same problem!
  • \(V=R^4\)
  • v₁ through v₄ as the standard basis. or not??
• dammit. shoulda seen that. thought that they were elements in the vectors.

• 2.B problem 8
• direct sum means intersection is zero,
• so when u add them together, u can just show it is linearly independent
• 0 is in the set of any span of vecs.
  • there is always a linear combo to 0! you need to show it's the only one.

• 2.C 1
• does the strat of just going back and looking for stuff work?
  • jana, says, yeah! pretty solid.
• use the appropriate results and qoute them clearly.
• how to include the actual relevant info:
  • not in parentheticals?
• how to make stuff not italic!

\DeclareMathOperator{\span}{span}

• 2.C 3
• prove:
  • R³ is a subspace of R³!
    • reverse double containment
• planes that pass through origin in R³ are not = R²,
  • they are isomorphic!

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them.

• this doesnt make them the same thing, but u can do the property mapping thing
• to take props from R² to R³, we need to know more about isomorphic
• to show a subspace:
  • additive iden
  • SCAMUL
  • addition
  • closed
  • and, also, it needs to live in the parent space.

• 2.C 4
• like karen's but with polynomials

fin. until thursday.

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