Linear Algebra

Linear Algebra- It includes the theory and applications of linear systems of equations, linear transformations and eigen value problems.

Matrix- Matrix is known as rectangular arrays of numbers. They are useful because they enable us to consider an array of many numbers as a single object and help us to perform calculations with these single object in a very compact form. 'A mathematical 'Shorthand'.

Order of a matrix- (m×n) matrix : m- rows and n- columns.

Square matrix- If m = n, then A is n×n square matrix.

Diagonal(main)/ principal diagonal- The diagonal containing a_11, a_{22, … ,} a_nn.

Rectangular matrix- A matrix which is not square is known as rectangular matrix.

Vectors- Row vector : matrix having one row (1×n)
                Column vector : matrix having one column (m×1)

Transposition- Interchanging row and columns. If A is m×n matrix [a_ij] then transpose of A, A^T is  n×m matrix [a_ji].

Symmetric matrix- A square matrix : A^T = A.

Skew-Symmetric matrix- A square matrix : A^T = -A.

Equality of matrix- Two matrices A = [a_ij] and B = [b_ij] are equal. i.e., A = B
if-

1.  They are of same order.
2. The corresponding entries are equal, a_ij = b_ij ∀ i, j

Matrix addition- The addition of two matrices

A = [a_ij] and B = [b_ij] is A + B,

if-

1. They are of same order.
2. A + B = [a_ij + b_ij], i.e., additing corresponding entries.

Scalar multiplication- The product of any matrix A = [a_ij] by a scalar c is cA = c[a_ij] = [ca_ij], i.e., multiplying each by c.

Zero matrix- A matrix with all entries Zero is called Zero matrix.