Midterm 1 Flashcards
What is a linear equation?
An equation in the form a1x1+a2x2+…+anxn=b where a1…an and b are all real numbers and are not ALL zeroes.
What is a system of linear equations?
A finite set of linear equations with the same number of n variables.
What does it mean for a linear system to be consistent?
There exists a solution.
What does it mean for a linear system to be inconsistent?
There exists no solution.
What is a homogenous linear equation?
A linear equation in which the constant terms equal zero. a1x1+a2x2+…+anxn=0.
What is a homogenous linear system?
A linear system in which the constant terms of each equation are zero. a11x1+a12x2+...a1nxn=0 a21x1+a22x2+...a2nxn=0 . . .
When are there infinitely many solutions to a linear system?
When there is at least 1 free variable.
When is there one unique solution to a linear system.
When there are no free variables.
When is there no solution to a linear system?
When the system is inconsistent.
What is the solution set to a single linear equation in two variables?
A line.
What is the solution set to a single linear equation in three variables?
A plane.
What is required for the consistency of an augmented matrix?
The rightmost column of the augmented matrix must not be a pivot column. And there must not exist a row where [0…0 | b] where b is not equal to zero.
True or false? A homogenous system is always consistent because the rightmost column always remains zero.
True.
What is a trivial solution and how does it tie into a homogenous system?
A trivial solution is x1=0,x2=2,…,xn=0. A trivial solution is a solution to a homogenous system.
What is the free variable theorem for homogenous systems?
If a homogenous system has n variables and its reduced row echelon form has r non-zero rows then the amount of free variables in that homogenous system is simple (n-r) free variables.
What is the second theorem for homogenous systems regarding solutions?
If there are more unknowns than equations then the system has infinitely many solutions. I.e. 5 variables with 4 equations.
What is a zero vector?
A vector with all components equal to 0.
When are the two vectors A and B equal?
When all the components that are in the same position in vector A and vector B are equal.
What is the parallelogram rule?
When you add vectors u and v the sum is the diagonal of the parallelogram formed by the two vectors.
When can you add two matrices?
When they both have the same size.
What is the transpose of a matrix?
It is the matrix formed by exchanging the rows and columns of a matrix. I.e. the row 135 in matrix A becomes the column 135 in the transpose of matrix A.
Theorem regarding equivalence and inverses of matrices?
If B and C are inverses of Matrix A then B=C.
What is (AB)^-1?
B^-1A^-1. I.e. inverse putting on socks and then shoes is removing shoes and then removing socks.
For a square matrix A which is mxm, what is A^0?
The identity matrix with dimensions mxm.
For a square matrix A which is mxm, what is A^n?
AxAxAxA….n times.
For a square matrix A which is mxm, what is A^-n?
A^-1 x A^-1 x A^-1…n times.
What is the matrix polynomial of the following polynomial? 3-2x-4x^2?
3I-2A-4A^2.
