Linear Algebra

Question 1

A subspace of Rn is any collection S of vectors in Rn such that:

Answer 1

(i) The zero vector 0 is in S.
(ii) If u and v are in S, then u + v are in S. (S is closed under addition.)
(iii) If u is in S and c is a scalar, then cu is in S. (S is closed under scalar multiplication

Question 2

A basis for a subspace S of Rn is a set of vectors in Rn that:

Answer 2

(i) spans S and
(ii) is linearly independent.

Question 3

Let A be an m × n matrix. The null space of A is…

Answer 3

The subspace of Rn consists of
solutions to the homogeneous linear system Ax = 0

Question 4

True or False.
Using that 1/|z| is a positive real number for z ̸= 0.

Answer 4

True. Explain

Question 5

True or False.
Matrix Multiplication is a row by column operation.

Answer 5

True. Explain

Question 6

True or False
If A is a square matrix, then A 1 AT is a symmetric matrix.

Answer 6

True. Explain

Question 7

True or False.
For any matrix A, AAT and ATA are symmetric matrices.

Answer 7

True. Explain

Question 8

If A is an invertible n x n matrix, then the system of linear equations given by Ax = b has the unique solution x = A-1b for any b in Rn.

Answer 8

True. Explain

Question 9

Let A be a matrix whose entries are real numbers. For any system of linear equations Ax b, exactly one of the following is true:
a. There is no solution.
b. There is a unique solution.
c. There are infinitely many solutions.

Answer 9

True. Explain

Question 10

A transformation T: Rn S Rm is called a linear transformation if
1. T(u + v) =T(u) + T(v) for all u and v in Rn and
2. T(cv) = cT(v) for all v in Rn and all scalars c.

Answer 10

True

Question 11

Elementary Row Operations has 3 steps name them.

Answer 11

1) Interchange Rows
2) Mutiltply a row
3) Add a multiple of a row to another

Question 12

True or False.
A system of linear equations with augmented matrix [A|B] is consistent if and only if b is a linear combination

Answer 12

True

Question 13

True or False.
A square matrix is symmetrical if AT = A that is if A is equal to its own transpose

Answer 13

True

Question 14

How do you find the inverse of a 2 x 2 matrix and verify it? List the steps?

Answer 14

1) Find the determinant
2) Multiply the inverse by the determinant
3) Verify by multiplying the original matrix A by the newly found inverse.

Question 15

Solving systems using the inverse.
X = A-1b. Name each part of the proof.

Answer 15

X = The coeffecients
A-1 = the inverse
b = Solutions of the systems

Question 16

True or False.
If A is an invertible matrix so is the AT and (AT)-1 = (A-1)T

Answer 16

True

Question 17

Steps to find the corresponding eigenvectors with a given eigenvalue.

Answer 17

1) Find the null space of matrix A [A - eigenvalue(identity)
2) Row reduce
3) Get leading 1’s
4) Use the free variable make equal to t or s
5) Find the span

Question 18

Determine the eigenvalues of following matrix. Analyze the steps.

Answer 18

1) Det(A - lambda(I)) = Multiply each diagonal by (-lambda)
2) Use cofactor expansion
3) Calculate each determinate
4) Simplify and solve for each eigenvalue

Question 19

Determine the eigenvectors from the eigenvalues found. State the steps.

Answer 19

1) Row reduce and find the leading 1’s
2) Then find the span

Question 20

True or False.
Every 2 x 2 with eigenvalues 0 and 1 is diagonalizable.

Answer 20

True. If the 2 x 2 matrix has 2 distinct eigenvalues, then it is diagonalizable

Question 21

True or False.
Every 3 x 3 martix with eigenvalues 0 and 1 is invertible.

Answer 21

False. It has an eigenvalue an eigenvalue of 0

Question 22

True or False. If you have an eigenvalue of 0 then your matrix is invertible.

Answer 22

False. If you have an eigenvalue of 0 then your 3 x 3 matrix is not invertible.

Question 23

True or False.
If a is a square matrix and det(A) /= ), then the rows of A are linearly independent

Answer 23

True.

Question 24

True or False.
Every upper-triangle square matrix is diagonalizable.

Answer 24

False.

Question 25

True or False.
Every upper triangular matrix does not have any complex eigenvalues.

Answer 25

True.

Question 26

How do you know that the det = 0 of a matrix A without finding the actual determinant?

Answer 26

1) If a matrix is linearly independent is non zero
2) If a matrix is linearly dependent than the det = 0

Question 27

True or False.
The columns of a m x n matrix form an orthonormal set if and only if QTQ = I

Answer 27

True.

Question 28

A square matrix Q is orthogonal if and only if Q-1 = QT

Answer 28

True

Question 29

What are we doing when we normalize orthogonal vectors?

Answer 29

We are turning them into unit vectors.

Question 30

True or False.
If A is a symmetric matrix than the eigenvalues of a are real.

Answer 30

True.

Question 31

True or False.
For any n x n matrix A, A and the rref(A) will have the same determinant

Answer 31

False.

Question 32

True or False.
For any n x n matrix A, A nd rref(A) will have the same eigenvectors

Answer 32

False