02 Vector Spaces
Set¶
collection of well-defined elements
Vector Space¶
A non-empty set \(V\) with binary operations \(\oplus\) and \(\odot\), which satisfies the following rules
| Law | |
|---|---|
| Closure Law wrt Addition | \(\forall u,v \in V, \quad \vec u \oplus \vec v \in V\) |
| Commutative Law wrt Addition | \(\vec u \oplus v = \vec v \oplus u\) |
| Associative Law wrt Addition | \((\vec u \oplus \vec v) \oplus \vec w = \vec u \oplus (\vec v \oplus \vec w)\) |
| Existence of additive identity | For \(\vec u \in V\), there exists \(\vec 0 \in V\) such that \(\vec 0 \oplus \vec u = \vec u \oplus \vec 0 = \vec u\) \(\vec 0\) is not necessarily \((0, 0)\) |
| Existence of additive inverse | For \(\vec u \in V\), there exists \(-u \in V\) such that \(\vec u \oplus (- \vec u) = (-\vec u) \oplus \vec u = \vec 0\) |
| Closure Law wrt multiplication | For any scalar \(\alpha\) (any real no) and \(\vec u \in V\) \(\alpha \odot \vec u \in V\) |
| Distributive Law (Right-Side) | For \(u, v \in V\) and scalar \(\alpha\) \(\alpha \odot (\vec u \oplus \vec v) = (\alpha \odot \vec u) \oplus (\alpha \odot \vec v)\) |
| Distributive Law (Left-Side) | For \(u \in V\) and scalars \(\alpha, \beta\) \((\alpha + \beta) \odot \vec u = (\alpha \odot \vec u ) \oplus (\beta \odot \vec u)\) |
| Distributive Law (Variation) | For \(u \in V\) and scalars \(\alpha, \beta\) \((\alpha \beta) \odot \vec u = \alpha \odot (\beta \odot \vec u)\) |
| Existence of unity | For \(u \in V\) \(1 \odot \vec u = \vec u\) |
Known Vector Spaces¶
- Real numbers
- \(R_2, R_3, R_n\)
- matrices
- polynomials
- form \(ax^n + bx^{n-1} + \dots + \alpha, \quad a, b \in R, \quad n \in Z\)
- \(P_n\) means degree of the polynomial \(\le n\)
- continuous functions
Subspace¶
Let \(S \subset V\) vector space. Then, \(S\) is a subspace if
- \(\vec 0 \in S\)
- \(\forall u, v \in S, \quad u \oplus v \in S\)
- \(\forall u \in S, \quad \alpha \odot u \in S\)
Trick to identify is if sum of powers of multiplicative terms is 1 For eg, \(x^a y^b + w^c z^d\) is subspace if \(a + b= 1, c + d = 1\)
Polynomial¶
\(P_n\) is a polynomial where degree \(\le n\)
For eg, even \((1 + x)\) is \(P_3\), as degree \(= 1 \le 3\)