Linear Algebra

Every system of linear equations has either NO solutions, 1 solution, or an INFINITE number of solutions

A system of equations (linear or not - over any space) is CONSISTENT if it is a NON-EMPTY set

A system of equations (linear or not - over any space) is INCONSISTENT if it is an EMPTY set

1. Algebraically eliminate variables

2. Simultaneous elimination

3. Gaussian elimination using matrices

1. First NONzero entry of 1st row is a 1 (leading 1)

• 2. Any rows consisting entirely of zeros are grouped together at the bottom on matrix
• 3. First NON zero entry of 2nd row is a 1 (occuring farther to the right than leading 1 in higher row

*4. Each column containing a leading 1 has zeros everyhwere else

A system of linear equations is said to be homogeneous if the constant terms are all zero; the system has the form:

Every HOMOGENEOUS system of linear equations is CONSISTENT (since all such systems have 0 as a solution. This is called the TRIVIAL solution

If there are any other solutions of a system of linear equations other than the trivial solutions, these are called NON-trivial solutions

A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions

A matrix is a rectangualr array of numbers

The numbers in the array are called entries

A column matrix is a matrix with only one column

A row matrix is a matrix with only one row

Two matrices are defined to be equal if they have the same size and their corresponding entries are equal

A matrix, all of whose entries are zero is called a zero matrix

1. A + 0 = 0 + A = A

2. A - A = 0

3. 0 - A = -A

4. A0 = 0; 0A = 0

A matrix with n rows and n columns is called a square matrix

A square matrix in whichn the entries outside the main diagonal are all zero



*tridiagonal matrices are also square matrices

A square matrix a, such that a_ij = 0 if i>j

A square matrix such that a_ij = 0 if i < j

• 

Matrix Addition is the operation of adding two matrices by adding the corresponding entries together.

*matrices of different sizes cannot be added (or subtracted)

Matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. The product c of matrices a and b is defined as:

c_ik = a_ij * b_jk

*the dimensions of the matrices must satisfy (mxn)(nxp) = (mxp)

The transpose of the (mxn) matrix a is the (nxm) matrix formed by interchaning the rows and columns such trhat row i becomes column i of the transposed matrix denoted by a^T

The trace of an n-by-n square matrix A - denoted by tr(A) - is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A

1. A + B = B + A (commutative law of addition)

2. A + (B + C) = (A + B) + C (associative law of addition)

3. A(BC) = AB + AC (associative law of multiplication)

4. A(B + C) = AB + CA / A(B - C) = AB - CA(left distributive law)

5. (B + C)A = BA + CA / (B - C)A = BA - CA(right distributive law)

If R is the rref of an nxn matrix A, then either R has a row of zeros or R is the identity matrix I

*Definiton - An identity matrix has the property that if A is a square matrix, then IA = AI = A

I_nxn = [a_ij] where a_ij = 1 if i = j, a_ij = 0 if i does not equal j

If A is a square matrix, and if a matrix B of the same size can be found such that AB = BA = I, then A is said to be invertible and B is called an inverse of A denoted by A^-1.

If no such matrix B can be found, then A is said to be singular (not invertible).

If B and C are both inverses of the matrix A, then B = C.

• Proof:
• Since B is an inverse of A, we have BA = I.
• Multiplying both sides on the right by C gives (BA)C = IC = C.
• But (BA)C = B(AC) = BI = B, so that C =B

• The matrix
• A = [ a b
• c d ]
• is invertible if ad - bc

• if A and B are invertible martices of the same size, then AB is invertible and
• (AB)^-1 = B^-1 A^-1

• Proof:
• If we can show that (AB)(B^-1A^-1) = (B^-1A^-1)(AB) = I,
• then we will have simultaneously shown that the matrix AB is invertible and
• that (AB)^-1 = B^-1 A^-1.
• But (AB)(B^-1A^-1) = A(BB^-1)A^-1 = AIA^-1 = AA^-1 =I.

*A similar argument shows that (B^-1A^-1)(AB) = I

A product of any number of invertible matrices is invertible, and the inverse of the product is the product of the inverses in the reverse order.