Linear Algebra Flashcards
Which two vectors are parallel?
Two vectors are parallel if one is a positive/negative scalar multiple of the other
Standard basis vectors
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).
When is a linear system consistent?
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/
What is a solution set of a linear system?
The solution set of a linear system is the set of all possible solutions for this set of linear equations.
A theorem about the linear system that has two solutions.
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.
A theorem about equivalent linear systems.
Two linear systems are equivalent if and only if they have the same number of equations ant the same solution set.
What is the difference between the echelon form of a matrix and a reduced echelon form?
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.
What is a vector space?
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.
Name 6 special matrices
Square matrix
Zero matrix
Diagonal matrix
Identity matrix
Upper triangular matrix (zeros in lower left corner)
Lower triangular matrix (zeros in top right corner)
What is an invertible matrix?
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.
What is a singular matrix?
A matrix that does not have an inverse is singular.
What is a property of an invertible matrix A?
Its inverse is always unique.
A theorem about a square invertible matrix and a linear system Ax = b. What can be said about its solutions?
There is one unique solution x = A^-1b
A theorem about a square singular matrix and a linear system Ax = b. What can be said about its solutions?
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.
What is a transpose of a matrix?
Transpose of a matrix is obtained by interchanging rows and columns.
A^t
What is a symmetric matrix?
A^t = A
So the matrix and its transpose are the same.
Name three useful properties of a transpose matrix.
(A^t)^t = A for any k*n matrix (A+B)^t = A^t + B^t (AB)^T = B^t*A^t
What is A^0?
I
What is A^r*A^s
A^r+s
(A^t)^t
A
(A+B)^t
A^t + B^t
(AB)^T
B^t*A^t
What is (A^r)^s
A^(r*s)
What are two ways to find an inverse of a matrix?
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)What is a linear subspace? (3)
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.
What is a homogeneous linear system?
If Ax = b and b = 0.
What is a null space?
Null space is the solution set of homogeneous linear system.
What is a row rank?
The maximum number of linearly independent rows in a matrix A is called the row rank of A,
What is a column rank?
The maximum number of linarly independent columns in A is called the column rank of A.
Is column rank bigger than row rank?
They are equal.
rank (A m*n) = ?
rank (A m*n) < = min (m, n)
When is a set of vectors S ={ v1, v2, … vp} is linearly dependent?
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.
A set of vectors S ={ v1, v2, … vp} is linearly independent if and only if…
a1v1 + a2v2 + … apvp = 0
implies that a1 = a2 = … ap = 0
What is the dimension of V (subspace of Rn)?
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).
When a matrix is invertible?
If A is a square matrix then it is invertable if and only if
- Null A = { 0 }
- Col A = R^n
- If det A is not 0
How to find a length of the vector?
Square root of the sum of squared elements.
How to find an angle between two vectors?
+ express in terms of the inner product of two vectors
cos THETA = (x1y1 + x2 * y2) / LxLy
+ express in terms of the inner product
x’y / √x’x*√y’y
What is the inner product of two vectors?
x’y = x1y1 + x2y2;
How to find a projection of x on y?
(x’y)*y / (y’y)
Draw a picture and prove it.
What is the length of a projection of x on y?
it is
Lx|cosTheta|= Lx|x’y / (√x’x√y’y )| = Lx|(x1y1 + x2 * y2) / LxLy|
What is an orthodonal matrix?
QQ’ = Q’Q = I
Properties ot determinant of a square matrix.
- det A = det A’
- If there are two identitical columns or two identical columns then det A = 0.
- If and only if det A is not equal to 0 then the matrix is invertable.
What happens to the det of A if multiple of one row is added to another one?
Nothing
What happens to the det of A if one row is multiplied with a (!=0)?
Nothing
What happens to the det of A if two roware interchanged?
determinant of the new matrix is equal to: - det A
WHat is the determinant of an inner product of two square matrices A and B?
det (AB) = det (A) det (B)
