Midterm 1

Question 1

What is a linear equation?

Answer 1

An equation in the form a1x1+a2x2+…+anxn=b where a1…an and b are all real numbers and are not ALL zeroes.

Question 2

What is a system of linear equations?

Answer 2

A finite set of linear equations with the same number of n variables.

Question 3

What does it mean for a linear system to be consistent?

Answer 3

There exists a solution.

Question 4

What does it mean for a linear system to be inconsistent?

Answer 4

There exists no solution.

Question 5

What is a homogenous linear equation?

Answer 5

A linear equation in which the constant terms equal zero. a1x1+a2x2+…+anxn=0.

Question 6

What is a homogenous linear system?

Answer 6

A linear system in which the constant terms of each equation are zero. 
a11x1+a12x2+...a1nxn=0
a21x1+a22x2+...a2nxn=0
.
.
.

Question 7

When are there infinitely many solutions to a linear system?

Answer 7

When there is at least 1 free variable.

Question 8

When is there one unique solution to a linear system.

Answer 8

When there are no free variables.

Question 9

When is there no solution to a linear system?

Answer 9

When the system is inconsistent.

Question 10

What is the solution set to a single linear equation in two variables?

Answer 10

A line.

Question 11

What is the solution set to a single linear equation in three variables?

Answer 11

A plane.

Question 12

What is required for the consistency of an augmented matrix?

Answer 12

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.

Question 13

True or false? A homogenous system is always consistent because the rightmost column always remains zero.

Answer 13

True.

Question 14

What is a trivial solution and how does it tie into a homogenous system?

Answer 14

A trivial solution is x1=0,x2=2,…,xn=0. A trivial solution is a solution to a homogenous system.

Question 15

What is the free variable theorem for homogenous systems?

Answer 15

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.

Question 16

What is the second theorem for homogenous systems regarding solutions?

Answer 16

If there are more unknowns than equations then the system has infinitely many solutions. I.e. 5 variables with 4 equations.

Question 17

What is a zero vector?

Answer 17

A vector with all components equal to 0.

Question 18

When are the two vectors A and B equal?

Answer 18

When all the components that are in the same position in vector A and vector B are equal.

Question 19

What is the parallelogram rule?

Answer 19

When you add vectors u and v the sum is the diagonal of the parallelogram formed by the two vectors.

Question 20

When can you add two matrices?

Answer 20

When they both have the same size.

Question 21

What is the transpose of a matrix?

Answer 21

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.

Question 22

Theorem regarding equivalence and inverses of matrices?

Answer 22

If B and C are inverses of Matrix A then B=C.

Question 23

What is (AB)^-1?

Answer 23

B^-1A^-1. I.e. inverse putting on socks and then shoes is removing shoes and then removing socks.

Question 24

For a square matrix A which is mxm, what is A^0?

Answer 24

The identity matrix with dimensions mxm.

Question 25

For a square matrix A which is mxm, what is A^n?

Answer 25

AxAxAxA….n times.

Question 26

For a square matrix A which is mxm, what is A^-n?

Answer 26

A^-1 x A^-1 x A^-1…n times.

Question 27

What is the matrix polynomial of the following polynomial? 3-2x-4x^2?

Answer 27

3I-2A-4A^2.