**Matrix **- Combination of rows and columns

Check for Linear Dependence - R2 = R2 - 2R1,

**When one of the rows is all zeros it is linearly dependent**
**Span** - Linear combination of vectors

**Rank** - Linearly Independent set

Good Related Read -

Span
**Vector Space **- Space of vectors, collection of many vectors

If V,W belong to space,

**V+W also belongs to space, multiplied vector will lie in R Square**
If the

**determinant is non-zero, then the vectors are linearly independent**. Otherwise, they are linearly dependent

**Vector space properties**
- Commutative x+y = y+x
- Associative (x+y)+z = x+(y+z)
- Origin vector - Vector will all zeros, 0+x = x+0 = x
- Additive (Inverse) - For every X there exists -x such that x+(-x) = 0
- Distributivity of scalar sum, r(x+s) = rx+rs
- Distributivity of vector sum, r(x+s) = rx+rs
- Identity multiplication, 1*x = x

**Subspace**
Vector Space V, Subset W. W is called subspace of V

**Properties**
W is subspace in following conditions

- Zero vector belongs to W
- if u and v are vectors, u+v is in W (closure under +)
- if v is any vector in W, and c is any real number, c.v is in W

Subset S belongs to V can be represnted as linear combination

v = r1v1+ r2v2+... rkvk

v1,v2 distinct vectors from S, r belongs to R

**Basis **-

**Linearly Independent spanning set. Vector space is called basis if every vector in the vector space is a linear combination of set**. All basis for vector V same cardinality

**Null Space, Row Space, Column Space**
Let A be m x n matrix

**Null Space **- All solutions for Ax = 0, Null space of A, denoted by Null A, is set of all homogenous solution for Ax=0
**Row Space **- Subspace of R power N spanned by row vectors is called Row Space
**Column Space** - Subspace of R power N spanned by column vector is called Column Space

**Norms - Measure of length and magnitude**
- For (1,-1,2), L1 Norm = Absolute value = 1+1+2 = 4
- L1 - Same Angle
- L2 - Plane
- L3 - Sum of vectors in 3D space
- L2 norm (5,2) = 5*5+2*2 = 29
- L infinity - Max of (5,2) = 5

**Orthogonal** - Dot product equals Zero

**Orthogonality** - Linearly Independent, perpendicular will be linearly independent

**Orthogonal matrix will always have determinant +/-1**

**Happy Learning!!!**