Introduction to Linear Algebra - created from Mind Map

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Linear Systems

Matrices

Properties of Matrix Operations

Determinants

Inverse of Matrix

ax=b : a linear equation in one variable (x)

Definition 1: a linear in "n" variables has the form, a1x1 + a2x2 + … + anxn = b

A linear system of m linear equations in n variables can be solved by Gaussian elimination method

Number of solutions of a system of linear equations: consistent system >has exactly one solution >has infinitely many solution inconsistent system >has no solution

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

Square matrices, m=n

A = [aiji]

Matric that has only one row is called row matrix or row-vector

Matrix that has only one column is called column matrix or column vector

A square matrix for which every term except of the main diagonal is zero, is called diagonal matrix

A diagonal matrix A for which aij = c for i=j and aij =0 for i not equal to j is called scalar matrix

Two m x n matrix A and B are said to be equal if aij = bij, that is corresponding elements are equal

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

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

If A is m x n matrix, AT is n x m matrix, then AT is called transpose of A

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

Matrix multiplication is a noncommutative operation

Elementary row operations: 1. Interchange two rows. 2. Multiply a row by nonzero constant. 3. Add a multiple of a row to another row

product of AB is not equal to product of BA

Properties of Matrix addition and Scalar Multiplication

Distributive & Associative Laws

The n x n matrix consist of only 0 beside diagonal located (left to right) 1 is called identity matrix of order n

Properties of Transpose

if AT = A, matrix A is called symmetric

Row-Echelon form and reduced row-echelon form

1. Closure property : A + B is again an m x n matirx

2. Commutative property : A + B = B + A

3. Associative property : (A + B) + C = A + (B + C)

4. Associative property of multiplication : (cd)A = c(dA)

5. Distributive property: c(A + B) = cA + cB

6. Distributive property: (c + d)A = cA + dA

7. Multiplication Identity: IA = A

8. Additive identity: the m x n zero matrix has property of A + 0 = A

9. Additive inverse: the m x n matrix (-A) has property of A + (-A) = 0

10. If cA = 0, either c = 0 or A = 0

A(B + C) = AB + AC [left-hand distributive law]

(D + E)F = DF + EF [right-hand distributive law]

A(BC) = (AB)C [associative law]

(AT)T = A

(A + B)T = AT + BT

(AB)T = BTAT

(rA)T = rAT

Any rows consisting entirely of zeros occur at the bottom of the matrix

For each row that doesn't consist entirely of zeros, the first nonzero entry is 1 (called leading 1)

Gauss-Jordan elimination : both right upper and left lower triangle is reduced to 0

Gaussian elimination: only left lower triangle is reduced to 0

2 x 2 matrix, det(A) = ad - bc

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)

Properties of Determinants

1. det(A) = det(AT)

2. If matrix B results from matrix A by interchanging two rows (columns) of A then det(B) = -det(A)

3. If two rows (columns) of matrix A are equal, then det(A) = 0

4. If a row (column) of matrix A consists entirely of zeros, then det(A) = 0

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. 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. If 3 x 3 matrix A consists of 0 on entirely upper right triangle, then det(A) = (a11 x a22 x a33)

8. det(AB) = det(A)det(B)

The determinant Mij is called the minor of aij.

The cofactor of Aij is defined as Aij = (-1)^(i+j) x minor of aij

adjA : adjoint of A

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