# Key Concepts in Linear Algebra

I. Notation s: Vectors and Vector Spaces

• n-dimensional vector: Ordered n-tuple

• Notation:

where
(real-valued elements) is a 3-dim column vector. A row vector is denoted by

• Vector Space:

• n-dimensional Euclidean space

• i^{th} elementary vector :

•

II. Linear independence, linearly independent vectors, basis of a vector
space,

subspaces, dimension (rank) of a vector space, space spanned by a collection

of vectors

III. Inner (dot) Product of two vectors

• Other notations:

the
feature space.

• Length of a vector, vector norm

• Angle between two vectors,

• Mutually orthogonal vectors,

• Pythagoras’ theorem, Triangle inequality, Cauchy-Schwartz
inequality,

• Orthogonal basis, orthonormal basis.

• Coordinates of a vector in terms of an o.n.b.

• Gram-Schmidt process for forming an o.n.b., Extension of an o.n.b.

for a subspace V to an o.n.b. for the space V_{n .}

• Orthogonal Complement of a vector space

IV. Matrices (definition, sums and products, transpose

• Rank of a matrix

• Space spanned by columns of a matrix **A,**

• Null space of a matrix,

• Matrix norm

V. Trace, Determinant, Eigen–values and Eigen-vectors and their properties, Spectral decomposition of a real symmetric matrix A.

VI. Singular Value Decomposition of a matrix A (SVD).

VII. Simultaneous diagonalization of two matrices

VIII. Symmetric and Idempotent matrices, Partitioned matrices.

IX. Linear forms, Quadratic forms, Positive semi-definite (p.s.d) (n.n.d.), Positive Definite (p.d.).

X. Systems of Linear Equations (Solution space), generalized inverse of a matrix, computation of a g-inverse, Moore-Penrose inverse of a matrix.