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Introduction to Linear Algebra

Linear Systems
1. ax=b : a linear equation in one variable (x)
2. Definition 1: a linear in "n" variables has the form, a1x1 + a2x2 + … + anxn = b
3. A linear system of m linear equations in n variables can be solved by Gaussian elimination method
4. Number of solutions of a system of linear equations: consistent system >has exactly one solution >has infinitely many solution inconsistent system >has no solution
5. Operations that produce equivalent systems: 1. Interchange two equations 2. Multiply an equation by a nonzero constant 3. Add a multiple of an equation to another equation
Matrices
1. Square matrices, m=n
2. A = [aiji]
3. Matric that has only one row is called row matrix or row-vector
4. Matrix that has only one column is called column matrix or column vector
5. A square matrix for which every term except of the main diagonal is zero, is called diagonal matrix
6. A diagonal matrix A for which aij = c for i=j and aij =0 for i not equal to j is called scalar matrix
7. Two m x n matrix A and B are said to be equal if aij = bij, that is corresponding elements are equal
8. If A and B are m x n matrices, then the sum of A and B is the m x n matrix C, defined by cij = aij + bij
9. If A is m x n matrix and r is real number, then the scalar multiple of A by r, rA, is m x n matrix B, then B = rA
10. If A is m x n matrix, AT is n x m matrix, then AT is called transpose of A
11. If A is m x p matrix, B is p x n matrix then product of A and B is the m x n matrix C
12. Matrix multiplication is a noncommutative operation
  1. product of AB is not equal to product of BA
13. Elementary row operations: 1. Interchange two rows. 2. Multiply a row by nonzero constant. 3. Add a multiple of a row to another row
Properties of Matrix Operations
1. Properties of Matrix addition and Scalar Multiplication
  1. 1. Closure property : A + B is again an m x n matirx
  2. 2. Commutative property : A + B = B + A
  3. 3. Associative property : (A + B) + C = A + (B + C)
  4. 4. Associative property of multiplication : (cd)A = c(dA)
  5. 5. Distributive property: c(A + B) = cA + cB
  6. 6. Distributive property: (c + d)A = cA + dA
  7. 7. Multiplication Identity: IA = A
  8. 8. Additive identity: the m x n zero matrix has property of A + 0 = A
  9. 9. Additive inverse: the m x n matrix (-A) has property of A + (-A) = 0
  10. 10. If cA = 0, either c = 0 or A = 0
2. Distributive & Associative Laws
  1. A(B + C) = AB + AC [left-hand distributive law]
  2. (D + E)F = DF + EF [right-hand distributive law]
  3. A(BC) = (AB)C [associative law]
3. The n x n matrix consist of only 0 beside diagonal located (left to right) 1 is called identity matrix of order n
4. Properties of Transpose
  1. (AT)T = A
  2. (A + B)T = AT + BT
  3. (AB)T = BTAT
  4. (rA)T = rAT
5. if AT = A, matrix A is called symmetric
6. Row-Echelon form and reduced row-echelon form
  1. Any rows consisting entirely of zeros occur at the bottom of the matrix
  2. For each row that doesn't consist entirely of zeros, the first nonzero entry is 1 (called leading 1)
  3. Gauss-Jordan elimination : both right upper and left lower triangle is reduced to 0
  4. Gaussian elimination: only left lower triangle is reduced to 0
Determinants
1. 2 x 2 matrix, det(A) = ad - bc
2. 3 x 3 matrix, det(A) = (a11 x a22 x a33) + (a12 x a23 x a31) + (a13 x a21 x a32) - (a13 x a22 x a31) - (a11 x a23 x a32) - (a12 x a21 x a33)
3. Properties of Determinants
  1. 1. det(A) = det(AT)
  2. 2. If matrix B results from matrix A by interchanging two rows (columns) of A then det(B) = -det(A)
  3. 3. If two rows (columns) of matrix A are equal, then det(A) = 0
  4. 4. If a row (column) of matrix A consists entirely of zeros, then det(A) = 0
  5. 5. If B is obtained from A by multiplying any row (column) of A by a real number c, then det(B) = c[det(A)]
  6. 6. If B is obtained from A by adding to each element of the r-th row (column) of A a constant c times the corresponding elementh of the s-th row (column), r not equal to s, then det(B) = det(A)
  7. 7. If 3 x 3 matrix A consists of 0 on entirely upper right triangle, then det(A) = (a11 x a22 x a33)
  8. 8. det(AB) = det(A)det(B)
Inverse of Matrix
1. The determinant Mij is called the minor of aij.
2. The cofactor of Aij is defined as Aij = (-1)^(i+j) x minor of aij
3. adjA : adjoint of A
4. An n x n matrix A is called nonsingular or invertible if there exists an n x n matrix B such that AB = BA = I(n) which matrix B is called as an inverse of A and shall be written as B = A^-1

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