Final Exam

Question 1

How can you tell if a T(x) is a linear transformation?

Answer 1

T(0vector) = 0vecter

Question 2

What is the domain D(T)?

Answer 2

• number of columns of matrix A
• input (pre-image) of the linear transformation

Question 3

What is the Cod(T)

Answer 3

• R^n space of the image
• R^n space (number of rows) of the matrix A

Question 4

What is R(T)?

Answer 4

columns space of A

R(T) = Span{v1, v2,…,vn}

the column vectors from the domain (input) of the transformation

Question 5

How can you determine if a matrix is onto?

Answer 5

1) Put the matrix in REF or RREF
2) Does the matrix have a pivot in every ROW?
If yes, the matrix is onto,
if not, it’s not onto

Question 6

How can you determine if a matrix is one-to-one?

Answer 6

1) Put the matrix in REF or RREF
2) Does the matrix have a pivot in every COLUMN? If yes, the matrix is one-to-one, if not, it’s not one-to-one

Question 7

What is an elementary row matrix? (notes 3.2)

Answer 7

If we perform a single elementary row operation on an identity matrix (I), then the result is matrix is called an elementary matrix (E)

Question 8

What is a property of the elementary row matrix?

Answer 8

If A is any matrix and E is an elementary matrix, then EA is the matrix obtained by performing the corresponding elementary row operation on A.

Question 9

What is A^T?

Answer 9

A^T is the transpose of A; changing rows to columns of A

Question 10

What are the operation laws for the transpose matrix?

Answer 10

(A+B)^T = A^T + B^T
(cA)^T = cA^T
(AB)^T = (B^T)(A^T)

IF A = A^T, A is a symmetric matrix

Question 11

How can you tell if a linear transformation is invertible?

Answer 11

If the linear transformation is both one-to-one and onto, then it is invertible

Question 12

How can you quickly find the inverse of a 2x2 matrix?

Answer 12

For a 2x2 matrix whose elements are [a b; c d], A^-1 = 1/(ad - bc) * [d -b; -c a]

Question 13

Mathematically, what is an inverse linear transformation?

Answer 13

T^-1(ȳ) = x̄ if T(x̄) = ȳ

Question 14

What are the operation laws for invertible matrices?

Answer 14

1. A^−1 is invertible, with (A^−1)^−1 = A
2. AB is invertible, with (AB)^−1 = (B^−1)(A^−1)
3. If AC = AD the C = D.
4. If AC = 0nxm then C = 0nxm

Question 15

What is a property of the inverse matrix?

Answer 15

For a linear system Ax=b, if A is invertible, then we can get a unique solution x = (A^1)y

Question 16

True or False; Any null-space is a subspace.

Answer 16

True, any null-space is a subspace.

Question 17

What are two quick ways to identify a subspace?

Answer 17

1) a homogenous (null space) set of equations
2) any span is a subspace

Question 18

How can you find a basis for a given set of vectors?

Answer 18

1) make the vectors into ROWS of a matrix, A
2) Put the matrix in echelon form
3) the non-zero rows correspond to the original vectors (columns) that form the basis

NOTE: you can do the same thing by leaving the vectors as columns for a matrix A, get the echelon form, but the pivot columns will be linearly independent and the
other columns will be linearly dependent on the pivot columns; the pivot columns correspond to the original columns that form a basis.

Question 19

What is the rank of a matrix?

Answer 19

After finding the basis of a matrix, you can find the rank.

Rank is the number of vectors in the basis. If there are 2 vectors in the basis, rank(A) = 2

Question 20

How do you find the basis of a row or column space?

Answer 20

1) Given a matrix (A), put the matrix in echelon form.
2) The non-zero rows correspond to the original rows in A which form the basis of the row space (don’t forget to write these as columns - vectors)
3) pivot columns correspond to the original columns in A that form a basis of the column space

Question 21

What is the null space and nullity?

Answer 21

Null space are the basis vectors of a homogenous set equations

Nullity is the number of vectors that form the null space

Question 22

What is the rank-nullity theorem?

Answer 22

rank(A) + nullity(A) = # of columns of A

Question 23

Given a matrix (A) and a set of vectors, how can you tell if the vectors are eigenvectors of A?

Answer 23

Multiply A by each vector. If the result is c*(the given vector) with the given vector identical to the original vector, then it is an eigenvector and c is the eigenvalue.

Question 24

When is A diagonizable?

Answer 24

An n × n matrix A is diagonalizable if and only if there are n independent eigenvectors

Question 25

If A is triangular (upper, lower, diagonal)matrix what does this tell you about it’s eigenvalues?

Answer 25

The eigenvalues of A are diagonal entries.

Question 26

True or False: An Invertible matrix is not necessarily diagonalizable, and a diagonalizable matrix is not necessarily invertible.

Answer 26

True