What is a Span{x₁, x₂, ..., x_k}?

the collection of ALL vectors that can be written in the form:

c₁v_{1 +} c₂v₂ + ... + c_kv_k

with scalers c_{1, ...,} c_p

what are the 3 Elementary Row Operations?

1. Replace a row by the multiple of another row added to it by a non zero number.
2. Multiply a row by a constant nonzero number
3. Switch the position of two rows.

What are the 8 Algebraic properties of Rⁿ?

1. x + y = y + x (Commutativity)
2. (x + y) + z = x + (y + z) (Additive Associativity)
3. α(βx) = (αβ)x  (Multiplicative Associativity)
4. 1 * x = x  (Multiplicative Identity)
5. x + 0 = 0 + x = x  (Additive Identity)
6. x + y = y + x = 0 iff y=-x  (Additive Inverse)
   
7. α(x + y) = αx + αy  (Scalar Distribution)
   
8. (α+ β)x = αx + βx  (Vector Distribution)

What is a homogeneous system?

A system of equations that can be written in the form Ax = 0, where A is a matrix n x m and x and 0 are vectors.

What is Nul A?

(Null Space)

the set of all x in Rⁿ and Ax = 0

What is a basis?

A Linearly Independent set in H that also spans H.

Define a Subspace?

any set H in Rⁿ that:

• The zero vector is in H
• For each u and v in H u + v is in H
• For each u in H and each scaler c the vector cu is in H.

What is the dimension of a subspace?

The number of vectors in any basis for the subspace (non 0)

What is the dimension of a vector space?

the number of vectors in a basis

What is the rank of a matrix?

the dimension of column space A

( the number of pivots in a matrix A)

What is Col A?

the span{a₁, ..., a_n)

(the span of the columns of a)

State the Rank Theorem

rank A + dim A = n columns

What is the inverse of A?

A•A^-1= I    and   A^-1•A=I

State the Basis Theorem

Let H be a p-dimensional subspace of Rⁿ. Any linearly Independent set of exactly p elements in H is automatically a basis for H as well as any set of p elements of H the spans H is automatically a basis for H.

State Cramers Rule

What is the det(AB)?

det (A) * det (B)

What is an eigenvector?

a nonzero vector x such that Ax = λx for some scaler λ. a scaler λ is called the eigenvalue of A. if there is a nontrivial solution x of Ax=λx ; such an x is called an eigenvector corresponding to λ. 

det (A^t) = ?

det (A)

If A is invertible then what is the det (A) = ?

det A is a non- zero number

A is invertible if and only if...

the number 0 is not an eigenvalue of A.

The determinant of A is not zero

State the characteristic equation

det(A-λI) = 0

(AB)^T= ?

B)

\

(AB)^T= B^TA^T= BA

What is the equation for solving a co-efficient matrix?

x = A^-1b

What is the adjoint of A?

The adjoint of A is the transpose of the matrix of cofactors and is denoted by adj(A).

What is Linear Indepenence?

the trivial solution is the only solution to Ax=0

What is an Orthogonal basis?

A basis that is an orthogonal set of unit vectors

What is orthogonal projection?

the vector y onto u s.t. y = y·u/(u·u) * u

y in W s.t y-y^ is orthogonal to W 

(y^= proj_wy)

Diagonolization

A has n linearly independent eigenvectors

(A is n x n matrix)

how do you diagonalize A=?

p=columns are e'vectors of A

D = P^-1AP is diagonal

What is the Best Approx Theory?

if w has an orthogonal basis {w_1, w_2, ..., w_p}; x=Rⁿ, then the closest vector in w to x is the vector:

x·w₁/(w₁·w_{1)w1 + x·w2/(w2·w2) + ...}

what is an orthogonal set?

a set of vectors whose dot product is 0

what is the dot product of x·y?

x₁y₁ + x₂y₂ + x₃y₃

what is the length formula?

the scaler || x || = (x·x)^1/2 = (x,x)^1/2

What is the definition of similarity?

if A and B are similar their characteristic equations are the same and thusly the same e'values

( A = P B P^-1)

what is the least square solution formula?

Ax=b -- A^T Ax = A^T b, represents the closest to the solution.

What does Gram-Schmidt do?

turns a basis into an orthogonal basis

eq. u1 = x1

u2 =x2 - x2·u1/(u1·u1)u1

u3 = x3 - x3·u1/(u1·u1) * u1 - x3·u2/(u2·u2) *u2 

what is the power method?

A x_o (x_o must have 1 as highest in the vector)

-> take out highest value for scaler new matrix b-> Ab = -> repeat...

then find what value it approaches