Linear Algebra Vocabulary/concepts

What is the dimension of a Matrix?

m rows x n columns
(Equations) x (Variables)

What is an Overdetermined Linear System?

More Equations than variables m>n

What is an Underdetermined Linear System?

More Variables than equations n>m

What is a Square Linear System?

Number of Variables is the same as Equations n=m

What is a consistent Matrix?

When a matrix has any solution, doesn't matter how many (can be infinite).

How can you tell if a system is independent?

Each line/plane should have a different slope

When is a system inconsistent?

When the system has no solution, ie when one or more rows has [ 0 0 | 1]

How can you tell if a system is dependent?

when each line/plane has the same slope.

What is a Type I elementary Row operation?

When you interchange 2 rows of a Matrix

What is a Type II elementary Row operation?

Multiplying a row by a non zero number

What is a Type III elementary Row operation?

Add a multiple of one row to another

When is a Matrix Upper triangular?

when all the lower triangle entries are 0, example
[1 2 3;
0 4 5;
0 0 6]

When is a Matrix Lower triangular?

when all the upper triangle entries are 0, example
[1 0 0;
2 3 0;
4 5 6]

What is a Diagonal matrix?

when both the upper and lower triangle is 0, example
[1 0 0;
0 2 0;
0 0 3]

What are row Equivalent Matrices?

Matrices A and B are row equivalent IFF matrix B can be obtained from matrix A using finite Elementary row operations

What is row echelon form of a matrix?

when the upper triangular form of a matrix has all diagonal pivots = 1, example
[1 2 3;
0 1 2;
0 0 1]

What is reduced row echelon form of a matrix?

when all pivot variables are 1 and eliminated from all other equations, example
[1 0 0 | 1
0 1 0 | 2
0 0 1 | 3]

What is a homogenous linear system? Is the System consistent?

Ax+b, but the vector b= 0 vector,
example
[1 2 | 0
3 4 | 0]
This system is ALWAYS consistent because A(0)=0

Theorem 2.2.1 properties

These are true for all scalars α and β and for any matrices A, B, and C

1. A+B = B+A
2. (A+B)+C = A+(B+C)
3. (AB)C = A(BC)
4. A(B+C) = AB+AC
5. (A+B)C =AC+BC
6. (αβ)A= α(βA)
7. α(AB) = A(αB)
8. (α+β)A=αA+βA(α+β)A=αA+βA
9. α(A+B)=αA+αB

What is the Identity matrix?

The identity Matrix is a diagonal Matrix where all diagonal entries are 1
example 3x3 matrix
[ 1 0 0;
0 1 0;
0 0 1]

What is the Transpose of a Matrix?

A transpose is where you swap the rows with their respective columns.
Example
A=
[1 2 3
4 5 6
7 8 9]

AT=
[1 4 7
2 5 8
3 6 9]

When is a Matrix Symmetric?

When A= AT
A=
[1 2 3
2 4 5
3 5 6]
AT=
[1 2 3
2 4 5
3 5 6]

When is a Matrix Antisymmetric?

When AT= -A
A=
[0 1 -2
-1 0 3
2 -3 0]
AT=
[0 -1 2
1 0 -3
-2 3 0]

What is a nonsingular Matrix?

A matrix has a multiplicative inverse.
ie AB= BA= I

What is a singular Matrix?

when a Matrix is NOT invertible

What is a degenerate matrix?

A square matrix that is not invertible.
ie determinant = 0

Theorem 2.3.1 Properties

for a square Matrix A ∈ R n×n all of these are equivalent:

1. all of the columns of A are pivot columns
   2 .The homogeneous linear system Ax=0 has a unique solution
2. A is a nonsingular matrix
3. A is a non-degenerate matrix
4. A is an invertible matrix

If A and B are non singular nxn (square) matrices, is the product AB nonsingular?

Yes AB will be nonsingular (invertible)

If matrix A is nonsingular is A transpose (A^T) nonsingular?

yes A^T will be nonsingular
A^T(A^−1)^T=(A^−1 A)^T
=I^T
=I

What order do elementary matrices go in?

RIGHT to Left in the order you did them
B= En*E2*E1*A

What are row equivalent matrices?

IF you have 2 matrices ( A and B), you will be able to multiply A by a finite number of elementary matrices to get B.

B= E1 * E2 *En*A

(since elementary matrices are invertible A is row equivalent to B) A= E1^-1 * E2^-1 * En^-1 *B

List the Properties of Theorem 2.4.1

(if A is a square Matrix all are equivalent)

1. All columns are pivot columns

2. The homogenous linear system (Ax=0) has a unique solution

3. A is nonsingular

4. A is nondegenerate

5. A is invertible

6. A is row equivalent to I

7. the rref of A is I

8. A is the product of elementary matrices

(more will be added later in course)

What is a set of valid inputs for a function called?

The Domain of a function

What are a set of valid outputs of a function called?

The Codomain of a function

True/False, A rule or set of rules defines EXACTLY ONE output in the codomain for EVERY input in the domain

True

How do you find the determinant of a 2x2 Matrix

A=
[ a b;
c d]
Det(A)= ad-bc

True/false, A matrix (nxn) has 2 equal rows, then the determinant is 0

True

True/false, Type III operations performed on a matrix CHANGES the determinant.

False

True/False, if a matrix (nxn) has a row of zeroes, then the determinant is 0

True

Is a matrix with determinant of 0 singular or nonsingular?

singular

Determinant of a matrix is equal to its determinant of its transpose, True/false?

True

What is the Adjugate Matrix?

The cofactor matrix Transposed,
eg
[a b
c d]

Adj=
[d -b
-c a]