Linear Algebra Flashcards
What is a matrix and what is it made of? What is the basic notation?
A rectangular array of numbers considered to be one mathematical object made up of elements.
Anm
n=rows
m=columns.
What is Aij?
Aij is the element of row i and column j. Can be numbers and/or variables.
What rules apply to addition and scalar multiplication?
To add matrices, they must be the same dimension.
The scalar is applied to each element to complete scalar multiplication.
What is a vector?
A vector can be thought of as a nx1 matrix. Can be any number of rows but always 1 column.
What is the inner product?
The inner product of a pair of vectors is the scalar. Multiply the elements of each vector then add the products to find the scalar.
What rules apply to matrix multiplication?
The number of columns of matrix A must be equal to the number of rows of matrix B.
The remaining outer dimensions create the dimension of the new matrix.
Eg. A 2x3 matrix and a 3x2 matrix can be multiplied to end up with a 2x2 matrix.
List the five special matrices
1) Square matrix
2) Symmetric matrix
3) Independent matrix
4) Identity matrix
5) Diagonal matrix
What defines a square matrix?
Transpose A to become A’ meaning the rows of A become the columns of A’.
A=A’
What defines a symmetric matrix?
It must be square. Transpose the matrix to find A=A’.
What defines a diagonal matrix?
Non-zero entries occur only on the main diagonal aka whole numbers diagonally and zero everywhere else.
What defines a identity matrix?
The matrix has 1’s is the diagonal as opposed to 0’s. The formula is IA=A.
What defines an independant matrix?
It is multiplied by itself and nothing changes.
What is Ax=b? What are we trying to find when we use this equation?
The linear equation of the matrix where A=matrix, x=vector, and b=vector.
Trying to find if a unique solution exists.
What is A^-1b=x?
It is the inverse of Ax=b and is used to find is a unique solution exists. We use the inverse because we cannot divide matrices.
What is a determinant used for? What does is A=0 mean?
To determine if a matrix (A) has an inverse (A^-1). If A=0 then A^-1 does not exist.
List 2 things about the submatrix
1) The submatrix is Aij
2) Found by removing rows and columns of A to get Aij
What is the cofator?
The hidden “+/-“ associated with the matrix.
What are the steps to Cramer’s Rule?
1) Find determinant of A
2) Construct ancillary matrix
3) Replace vector B throughout the columns of matrix A and find determinant
4) Take the det of each new matrix and divide by det A to find the solutions for vector X
What do A=0 and A ‘does not equal’ 0 indicate about the matrix?
If A=0 then A in singular and the inverse does not exist.
If A does not equal 0 then A is nonsingular and the inverse does exist.
What are the steps for row echelon decompostion?
1) Write the augmented matrix
2) Perform elementary row operations such that all the elements below the diagonal are zero
3) Resulting matrix is R
What options exist for elementary row operations?
1) Swap two rows
2) Multiply a row by a nonzero real number
3) Add or subtract rows
What is the rank of the matrix? What does it mean if a matrix does not have full rank?
The number of linearly independent rows or columns. If a matrix is singular it does not have full rank.
What is the row echelon form and the reduced row echelon form?
The row echelon form contains zeros beneath the diagonal. The reduced row echelon form contains all ones on the diagonal and zeros everywhere else.
What is the first nonzero element called?
A pivot
What is the leontif model?
The model describes an economy with interlinked industries. It describes that output from one industry can become an input to another industry.
No answer req: Each industry faces an external/final demand in addition to supplying its own goods to other industries. The amount of each good is used to supply internal and external demands.
NA
In the equation X=Ax + b what does each variable represent?
A = the matrix of technical coefficients or inputs and outputs required per industry
x = the supply vector
b = the final demand vector
X = the quantity that needs to be produced
What are the steps to find X in the leontif matrix?
1) Complete reduced row echelon
2) Compute the identity matrix minus the technical coefficient matrix (I-A)
3) Find the leontif inverse (I-A)^-1 which gives X
4) Multiply x and b (the final demand vector)
When is the leontif model “employed”? What is a sensitivity analysis?
To predict ripple effects.
A sensitivity analysis looks at what happens to changes in the supply vector and demand vector as each increases or decreases.
What are Markov Chains?
Statistical model that describes a sequence of possible events and is applied when there are repeated observations made at regular time intervals.
NA: Matrix of probability - columns must add up to 1
NA
