Backlinks
#flo
1 Talking about the reading (vector spaces)
1.1 Vector space
1.1.1 Identity
- It would be the additive identity, because the multiplicitive one doesn't count because multiply doesn't take two elements from the same field #### Operations
- Scalar multiplication
- Not a multiplication on \(V\)
- We need another field of scalars
- Fundamental difference: operates on different objects (only happens on scalar multiplications)
- addition #### Linearity
- Something that's linear means "things work for addition and scalar multiplication"
- Take \(-2x+1y=3\)
- Multiplying by scalars
- adding them
- similar to a line in standard form–slope stays constant
- Take \(2x-3y+1z=2\)
- a plane in 3d
- if you pick a direction, the slope stays the same
- thus, a plane is linear #### Vector
- Something in a vector space
- inifinite lists
- It's like decimals, except you can chose any number instead of just [0-9]
- base infinity basically
- Most common vector space
- \(\mathbb{F}^n\), like \(\mathbb{R}^3\) (might also be \(\mathbb{C}^2\) or something, although that's hard to visualize)
- #definition canonical
- something "standard", basically everyone should know what you are talking about
- canonical vector space is \(\mathbb{R}^2\) #### Distributive property
- Important to tie operations together
1.1.2 Vector Space as a Set of Functions
- like \(\mathbb{R}^{[0, 1]}\): the functions from \([0, 1]\) that end
up as real numbers
- Identity = \(f(x) = 0\) #### Subspaces
- A subspace of this has to be a group on it's own
- Conditions for a subspace
- See 1.34
- Just check
- additive identity
- closed under addition
- closed under scalar multiplication
- What other subspaces of this vector space are there that also have a
domain from \([0, 1]\)?
- Like continuous functions from zero to one
- functions who's derivatives are continuous or constant or zero
- even functions are also a subspace KBe20math530srcEvenFunctionsAreSubspacesOfFtotheS.png
- Subspaces of \(\mathbb{F}^3\)
- Most contain infinite vectors (except \(\{ 0 \}\))
- \(\begin{bmatrix}x\\y\\0\end{bmatrix}\) is a subspace with infinite vectors #### Notation
- #note \(\mathbb{F}^2\) is almost always either \(\mathbb{R}^2\) or \(\mathbb{C}^2\), mostly \(\mathbb{R}^2\)
1.2 Direct sums
- Something that isn't a direct sum
- in \(\mathbb{R}^3\), \(\begin{bmatrix}x\\y\\0\end{bmatrix}\) and
\(\begin{bmatrix}x\\x\\0\end{bmatrix}\)
- Two ways to write \(0\):
- \(\begin{bmatrix}0\\0\\0\end{bmatrix} + \begin{bmatrix}0\\0\\0\\\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix} = \begin{bmatrix}1\\1\\0\end{bmatrix} + \begin{bmatrix}-1\\-1\\0\end{bmatrix}\) ## \(\mathbb{F}^\infty\)
- Two ways to write \(0\):
- in \(\mathbb{R}^3\), \(\begin{bmatrix}x\\y\\0\end{bmatrix}\) and
\(\begin{bmatrix}x\\x\\0\end{bmatrix}\)
- Functions from naturals to your field, (assign an element to each
natural)
- that would be the same as ordering the elements in your field?
- Tons of functions, any one is an infinite vector??
2 If and Only If proofs (iff)
- You have to take the proof in both directions
- Assumption: "now suppose the only way to write 0 as a sum of u1 +
… | um, where each uj is in Uj, is by taking each uj equal to 0"
- Assume the red part, then show the green part. Then, assume the green and show it gets the red.
- KBe20math530srcIfOnlyIfProofs.png
- #future geometrical interpretation of determinants
