Linear Algebra

Question 1

Which two vectors are parallel?

Answer 1

Two vectors are parallel if one is a positive/negative scalar multiple of the other

Question 2

Standard basis vectors

Answer 2

By invoking the definitions of vector addition and scalar multiplication, any vector x = (x 1, x 2) in R 2 can be written in terms of the standard basis vectors i (1, 0) and j (0, 1).

Question 3

When is a linear system consistent?

Answer 3

Linear system is consistent if it does not have a pivot in the last column.
And the set of linear equations has at least one solution/

Question 4

What is a solution set of a linear system?

Answer 4

The solution set of a linear system is the set of all possible solutions for this set of linear equations.

Question 5

A theorem about the linear system that has two solutions.

Answer 5

If a LS has at least two solutions it means it has an infinite number of solutions.

I am not sure about the proof. See page 6.

Question 6

A theorem about equivalent linear systems.

Answer 6

Two linear systems are equivalent if and only if they have the same number of equations ant the same solution set.

Question 7

What is the difference between the echelon form of a matrix and a reduced echelon form?

Answer 7

REF is the same as EF, but in the REF each pivot is equal to 1 and is the only non-zero entry in the table.
EF is not unique and REF is unique.

Question 8

What is a vector space?

Answer 8

Vector space is a collection of all vectors with a certain number of elements.

Vector space is a set, so it is closed under addition and scalar multiplication.

Question 9

Name 6 special matrices

Answer 9

Square matrix
Zero matrix
Diagonal matrix
Identity matrix
Upper triangular matrix (zeros in lower left corner)
Lower triangular matrix (zeros in top right corner)

Question 10

What is an invertible matrix?

Answer 10

A is an invertible matrix if there exists B
such that
AB = BA = I
Where I is the identity matrix of the same size as A.

Question 11

What is a singular matrix?

Answer 11

A matrix that does not have an inverse is singular.

Question 12

What is a property of an invertible matrix A?

Answer 12

Its inverse is always unique.

Question 13

A theorem about a square invertible matrix and a linear system Ax = b. What can be said about its solutions?

Answer 13

There is one unique solution x = A^-1b

Question 14

A theorem about a square singular matrix and a linear system Ax = b. What can be said about its solutions?

Answer 14

There is always exists at least one b for which Ax=b has no solutions.
If there is at least one solution to it, then there are infinite number of solutions.

Question 15

What is a transpose of a matrix?

Answer 15

Transpose of a matrix is obtained by interchanging rows and columns.
A^t

Question 16

What is a symmetric matrix?

Answer 16

A^t = A

So the matrix and its transpose are the same.

Question 17

Name three useful properties of a transpose matrix.

Answer 17

(A^t)^t = A for any k*n matrix
(A+B)^t = A^t + B^t
(AB)^T = B^t*A^t

Question 18

What is A^0?

Answer 18

I

Question 19

What is A^r*A^s

Answer 19

A^r+s

Question 20

(A^t)^t

Answer 20

A

Question 21

(A+B)^t

Answer 21

A^t + B^t

Question 22

(AB)^T

Answer 22

B^t*A^t

Question 23

What is (A^r)^s

Answer 23

A^(r*s)

Question 24

What are two ways to find an inverse of a matrix?

Answer 24

Start with (A| I ) and proceed to ( I | A^-1)
or 
if A = (a b)
         (c d)
use a formula  A^-1 =
1/          * (d -b)
(ad-bc)   (-c  a)

Question 25

What is a linear subspace? (3)

Answer 25

A subset D of R^n is called a linear subspace of R^n if it satisfies the following conditions 0 is in D for all U and V in D u+v is also in D for all alpha (real number) and V in D, we have alpha*V is in D.

Question 26

What is a homogeneous linear system?

Answer 26

If Ax = b and b = 0.

Question 27

What is a null space?

Answer 27

Null space is the solution set of homogeneous linear system.

Question 28

What is a row rank?

Answer 28

The maximum number of linearly independent rows in a matrix A is called the row rank of A,

Question 29

What is a column rank?

Answer 29

The maximum number of linarly independent columns in A is called the column rank of A.

Question 30

Is column rank bigger than row rank?

Answer 30

They are equal.

Question 31

rank (A m*n) = ?

Answer 31

rank (A m*n) < = min (m, n)

Question 32

When is a set of vectors S ={ v1, v2, ... vp} is linearly dependent?

Answer 32

if p=1 and v1 = 0; | or p>1 and one of the vectors in S is a linear combination of the vectors in S.

Question 33

A set of vectors S ={ v1, v2, ... vp} is linearly independent if and only if...

Answer 33

a1v1 + a2v2 + ... apvp = 0 | implies that a1 = a2 = ... ap = 0

Question 34

What is the dimension of V (subspace of Rn)?

Answer 34

A number of elements in the basis of V. The unique number of vectors in each basis for V is called the dimension of V and is denoted by dim(V).

Question 35

When a matrix is invertible?

Answer 35

If A is a square matrix then it is invertable if and only if 1. Null A = { 0 } 2. Col A = R^n 3. If det A is not 0

Question 36

How to find a length of the vector?

Answer 36

Square root of the sum of squared elements.

Question 37

How to find an angle between two vectors? | + express in terms of the inner product of two vectors

Answer 37

cos THETA = (x1*y1 + x2 * y2) / Lx*Ly + express in terms of the inner product x'y / √x'x*√y'y

Question 38

What is the inner product of two vectors?

Answer 38

x'y = x1*y1 + x2*y2;

Question 39

How to find a projection of x on y?

Answer 39

(x'y)*y / (y'y) Draw a picture and prove it.

Question 40

What is the length of a projection of x on y?

Answer 40

it is | Lx*|cosTheta|= Lx*|x'y / (√x'x*√y'y )| = Lx*|(x1*y1 + x2 * y2) / Lx*Ly|

Question 41

What is an orthodonal matrix?

Answer 41

Q*Q' = Q'*Q = I

Question 42

Properties ot determinant of a square matrix.

Answer 42

1. det A = det A' 2. If there are two identitical columns or two identical columns then det A = 0. 3. If and only if det A is not equal to 0 then the matrix is invertable.

Question 43

What happens to the det of A if multiple of one row is added to another one?

Answer 43

Nothing

Question 44

What happens to the det of A if one row is multiplied with a (!=0)?

Answer 44

Nothing

Question 45

What happens to the det of A if two roware interchanged?

Answer 45

determinant of the new matrix is equal to: - det A

Question 46

WHat is the determinant of an inner product of two square matrices A and B?

Answer 46

det (AB) = det (A) det (B)