Linear Algebra Final

Question 1

What is the definition of Row Echelon Form?

Answer 1

1) Any rows consisting entirely of zeros are at the bottom.
2) In each nonzero row, the first entry (called the leading entry) is in the column to the left of any leading entries below it.

Question 2

What is row equivalent matrices?

Answer 2

If there is a sequence of row operations that convert one matrix into the other. (If so the two matrices have the same set of solutions)

Question 3

What constitutes a linearly inconsistent system?

Answer 3

When there is no set of values for the unknows that satisfies all of the equations. (I.E. no solution) Graphical representation in 2D would be the lines never crossing.

Question 4

What is the Rank of a matrix (Rank(A))?

Answer 4

Is the number of nonzero rows in the given matrix A that is in Row Echelon form.

Question 5

What is the bound variable?

Answer 5

)

Question 6

What is a free variable?

Answer 6

Any variable that is not a bound variable

Question 7

What is the definition of row-reduced echelon form?

Answer 7

1) It is in Row Echelon From
2) The first nonzero entry in each row is 1
3) The first nonzero entry in each row is the only nonzero entry in its column

Question 8

What is the definition of a homogenous linear system?

Answer 8

A system of linear equations having matrix form Ax = O, where O represents a zero column matrix

Question 9

How to prove that a homogenous linear system has infinite solutions?

Answer 9

If this determinant of the system is zero, then the system has an infinite number of solutions. Or if the system has fewer equations then unknowns.

Question 10

When does Ax = b have a solution for all b in R^m if A is an mxn matrix?

Answer 10

Only when rank(A) = m

Question 11

What is the definition of span?

Answer 11

The span of the vectors (v1,v2,…vn) is the set of all linear combinations of the vector v1 through vn.

Question 12

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

Answer 12

A system is consistent if there is at least one set of values for the unknowns that satisfies each equation in the system.

Question 13

When is Ax = b consistent given A is an m x n matrix with column vectors v1, v2, … vn and b is a vector in R^m?

Answer 13

It is consistent if and only if b is in the span(v1, v2, … vn)

Question 14

What are the conditions for a matrix to be invertible?

Answer 14

1) Matrix must be a square matrix
2) Rank(A) = n
3) The linear system Ax = b has a unique solution for every b in R^n
4) The homogenous linear system Ax = 0 has only the trivial solution x = 0
5) The reduced echelon form of A is A_n ( Columns of A are linearly independents)

Question 15

A set of vectors W in R^m is called a subspace of R^m if?

Answer 15

Is a special collection of vectors that satisfies the following. Remember that a span is always a subspace.
1) The vector 0 is in W
2) If you add any 2 vectors in the set together, you get a vector also in the set.
3) If you multiply any vector in the set by any scalar you geta vector also in the set.

Question 16

What is a basis?

Answer 16

A basis is for a subspace (W) - it is a set of linearly independent vectors that span W.

Question 17

What is a subset?

Answer 17

Any collection of vectors

Question 18

What is the column space of A?

Answer 18

It is defined as all the linear combinations of the column vectors of A (also knows as the span (column vectors of A))

Question 19

What is the null space of A?

Answer 19

The null(A) is just solutions to Ax = 0. Or in other words the set of x in R^n where Ax = 0 where A is an m x n matrix. Null space is always a span. The vectors of the null space are always linearly independents so they are the basis for null space of A.

Question 20

What is Nullity of A?

Answer 20

Nullity(A) = dim(null(A))

Question 21

What is rank nullity theorem state?

Answer 21

rank(A) + nullity(A) = n

Question 22

What is an eigenvector and eigenvalue?

Answer 22

We call a nonzero vector x an eigenvector of A with corresponding eigenvalue if Ax = λx

Question 23

Is the matrix A - λI invertible?

Answer 23

NO

Question 24

What does it mean when a matrix is diagonalizable?

Answer 24

There must exist an invertible matrix P and a diagonal matrix D such that A = PDP^-1. It also is only diagonalizable if and only if for every eigenvalue λ of A there is a unique eigenvector…that is to say the algebraic multiplicity of λ is equal to the geometric multiplicity of λ.

Question 25

What are the necessary requirements for A and B to be similar?

Answer 25

a) det(A) = det(B)
b) A is invertible if B is invertible
c) rank(A) = rank(B)\
d) A and B have the same characteristic polynomial
e) A and B have the same eigenvalues

Question 26

How do you know if a linear system has only one unique solution?

Answer 26

In a set of linear simultaneous equations, a unique solution exists if and only if, (a) the number of unknowns and the number of equations are equal, (b) all equations are consistent, and (c) there is no linear dependence between any two or more equations, that is, all equations are independent.

Question 27

What is projection of y onto x equation?

Answer 27

proj_x ^y = ((xy)/(xx)) *x

Question 28

What does an orthogonal set tell you>

Answer 28

If the set (S) is an orthogonal set of nonzero vectors then vectors v1,v2…vn are linearly independent. If the set is an orthonormal set then the vectors v1,v1,…vn are linearly independent.

Question 29

When do you have an orthogonal basis?

Answer 29

The set S is an orthogonal basis for W if S is orthogonal and S is a basis for W.

Question 30

What is the fastest way to figure our the dimension that a vector set spans?

Answer 30

You can put the vectors into a matrix and row-reduce it and however many pivots you get that is the dimension of the span.