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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

• ith 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 Vn .
• 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.