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Linear Algebra > Linear Algebra > Flashcards

Linear Algebra Flashcards

1
Q

Which two vectors are parallel?

A

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

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2
Q

Standard basis vectors

A

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).

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3
Q

When is a linear system consistent?

A

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/

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4
Q

What is a solution set of a linear system?

A

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

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5
Q

A theorem about the linear system that has two solutions.

A

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.

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6
Q

A theorem about equivalent linear systems.

A

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

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7
Q

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

A

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.

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8
Q

What is a vector space?

A

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.

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9
Q

Name 6 special matrices

A

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

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10
Q

What is an invertible matrix?

A

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.

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11
Q

What is a singular matrix?

A

A matrix that does not have an inverse is singular.

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12
Q

What is a property of an invertible matrix A?

A

Its inverse is always unique.

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13
Q

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

A

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

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14
Q

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

A

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.

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15
Q

What is a transpose of a matrix?

A

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

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16
Q

What is a symmetric matrix?

A

A^t = A

So the matrix and its transpose are the same.

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17
Q

Name three useful properties of a transpose matrix.

A 
(A^t)^t = A for any k*n matrix
(A+B)^t = A^t + B^t
(AB)^T = B^t*A^t
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18
Q

What is A^0?

A

I

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19
Q

What is A^r*A^s

A

A^r+s

20
Q

(A^t)^t

A

A

21
Q

(A+B)^t

A

A^t + B^t

22
Q

(AB)^T

A

B^t*A^t

23
Q

What is (A^r)^s

A

A^(r*s)

24
Q

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

A 
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)
25
Q

What is a linear subspace? (3)

A

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.

26
Q

What is a homogeneous linear system?

A

If Ax = b and b = 0.

27
Q

What is a null space?

A

Null space is the solution set of homogeneous linear system.

28
Q

What is a row rank?

A

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

29
Q

What is a column rank?

A

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

30
Q

Is column rank bigger than row rank?

A

They are equal.

31
Q

rank (A m*n) = ?

A

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

32
Q

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

A

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.

33
Q

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

A

a1v1 + a2v2 + … apvp = 0

implies that a1 = a2 = … ap = 0

34
Q

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

A

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).

35
Q

When a matrix is invertible?

A

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
36
Q

How to find a length of the vector?

A

Square root of the sum of squared elements.

37
Q

How to find an angle between two vectors?

+ express in terms of the inner product of two vectors

A

cos THETA = (x1y1 + x2 * y2) / LxLy
+ express in terms of the inner product

x’y / √x’x*√y’y

38
Q

What is the inner product of two vectors?

A

x’y = x1y1 + x2y2;

39
Q

How to find a projection of x on y?

A

(x’y)*y / (y’y)

Draw a picture and prove it.

40
Q

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

A

it is

Lx|cosTheta|= Lx|x’y / (√x’x√y’y )| = Lx|(x1y1 + x2 * y2) / LxLy|

41
Q

What is an orthodonal matrix?

A

QQ’ = Q’Q = I

42
Q

Properties ot determinant of a square matrix.

A
  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.
43
Q

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

A

Nothing

44
Q

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

A

Nothing

45
Q

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

A

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

46
Q

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

A

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

Linear Algebra (1 decks)

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