det ≠ 0 - Equation Ax=b has a unique solution - Matrix has full rank

1. Linearly independent 2. Span

Final Exam Flashcards by alexia oltramare

Do elementary row operations change the solution set?

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When does a linear system have 0 solutions?

[0 0 0 1]
0x + 0y + … = 1

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When does a linear system have 1 solution?

leading 1s = # variables

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When does a linear system have infinite solution?

leading 1s < # variables
(paramterize)

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Symmetric matrix

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Span

set of all linear combinations

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Linearly dependent

if there exists scalars (not all equal to 0) such that:
x1v1 + x2v2 + … + xnvn = 0

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Linearly independent

if the only scalars are all 0 such that:
x1v1 + x2v2 + … + xnvn = 0

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Homogeneous linear system

Ax = 0
Sends to zero vector

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Non-homogeneous linear system

Ax = b
(translations of Ax = 0)

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Homogenous system:
- If vector (v) is a solution to Ax = 0, then scalar (k) multiplied by v (kv) is what?

Also a solution to Ax = 0

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Homogenous system:
- If vectors (v1, v2) are solutions to Ax = 0, then:
v1 + v2 is what?

v1 + v2 is also a solution to Ax = 0

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Homogenous system:
- Any linear combination of homogenous solution (Ax=0) is what?

Also a solution

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Systems:
- If vector (a) is a solution to Ax = 0
- If vector (c) is a solution to Ax = b
Then vector a+c is a solution to what?

homogenous solution + non-homogenous solution = non-homogenous solution

a+c is a solution to Ax=b

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Systems:
- If vectors (a, c) are solutions to Ax = b
Then vector a-c is a solution to what?

Vector a–c is a solution to Ax=0 (homogenous)

non-homo – non-homo = homo

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Linear transformation

T(v+w) = T(v)+T(w)
T(kv) = kT(v)

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Invertible matrix

det ≠ 0

Equation Ax=b has a unique solution
Matrix has full rank

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Can non-square matrices have inverses?

Yes, but only on one side

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Subspace

u+v in V
ku in V

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Basis

Linearly independent
Span

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How can you determine the dimension of a vector space?

elements in basis

UNIQUE

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If a vector space has dimension n, then n linearly independent vectors must be a ___

basis

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If a vector space V has dimension n, and collection of vectors span(V) then ___

vectors are a basis

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Coordinates of a vector space
- v = arbitrary vector in V
- Basis: {v1,v2…vn}

a1v1 + a2v2 + … + anvn = v

Collection of scalars [a1 a2 … an]

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Find the matrix that represents the transformation (T) from basis A to basis B

1. Compute image of each basis vector of A: T(a1), T(a2) ... T(an) 2. Re-write in terms of basis B: T(a1) = x1b1 + ... + xnbn 3. From matrix using coefficients (x1...xn) as columns

Characteristic polynomial

det(A – λI)

How to find eigenvectors given λ?

null(A – λI)

Fundamental formula relating rank & dim(Null)

rank(A) + dim(null(A)) = n

Rank

linearly independent rows of the matrix # leading 1s in RREF

If a matrix is (mxn) what is it's rank?

Whichever is smallest m or n

Nullspace

Solution to Ax=0

Column space

span of the columns of A

How to determine the column space of a matrix?

- RREF A → find the columns with leading 1s - Col(A) = span(columns with leading 1s in ORIGINAL matrix A) - Basis for Col(A) = {columns of original matrix A which correspond to leading 1s in RREF A}

Row space

span of the rows of A

How to determine the row space of a matrix?

- RREF A - Row(A) = span (RREF’s rows of A)

Do row operations preserve the row space?

Yes!

Do row operations preserve the column space?

No! But the dim(col(A)) yes

Determinant of a triangular matrix

product on main diagonal

If rank < n, then determinant = ?

det = 0

What is the minor? (Mij)

determinant submatrix remains after the ith row & jth column are deleted

What is the cofactor? (Cij)

Determinant properties: - Multiply all elements of 1 row/column by k

k * det

Determinant properties: If a row/column is a sum of two vectors

det can be written as the sum of the 2 determinants with each of the 2 row vectors and the remaining rows/columns as in the original

Determinant properties: If 2 rows/column are identical

det = 0

Determinant properties: Add to a row/column a multiple of another row

det stays SAME

Determinant properties: If 2 rows/column are linearly dependent

det = 0

det(AB) =

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

Eigenvector (v)

Av = λv v ≠ 0

Does changing the basis of a linear transformation change its characteristic polynomial?

The set of eigenvectors corresponding to a single eigenvalue together with the zero vector form a __

subspace of Rn

Eigenvectors corresponding to different eigenvalues are what?

Linearly independent

Diagonalisable if

1. dim(eigenspace) = multiplicity of eigenvalue for each λ 2. able to perform a change of coordinate that brings matrix to diagonal form D = P^-1 AP

Every eigenvalue must have at least how many eigenevctors?

If a charactertistic polynomial has n DIFFERENT eigenvalues (mult = 1) then?

diagonalizable

Complex #

z = a + bi

i^2 =

-1

Complex conjugate

z(bar) = a - bi

Moduluz: |z|

distance from point to the origin

Given an (nxn) matrix with REAL entries, if a+bi is an eigenvalue with eigenvector (v) then what is another eigenvalue/eigenvetcor?

Orthogonal

dot product = 0

Orthonormal

dot product = 0 length = 1

How to normalize a vector?

How to compute the length of a vector? ||v||

Orthogonal complement

set of vectors that are orthogonal to every vector in subspace H

A vector can be written as the sum of its orthogonal projection and what?

Orthogonal projection

v hat

Gram-Schmidt Orthogonalisation Proces

transform a basis → orthogonal basis

What does a least squares approximation do?

Solves the equation Ax=b as closely as possible least squares = vector y such that |b-Ay| is as small as possible...

2 methods for computing least squares approx

1. Find the projection b(hat) of b on the space spanned by the columns of A, then solve the equation Ax=b(hat). 2. Solve A^T Ax= A^T b

Line of best fit to n data points

Find the values of a,b in the linear equation y = a + bx that minimize the distance between the n data points

Orthonormal matrix

An orthonormal change of coordinate transforms a symmtetric matrix into what?

Into another symmetric matrix

A square symmetric matrix with real entries has __ eigenvalues

real eigenvalues

If v1 and v2 eigenvectors of A with different eigenvalues, then

v1 and v2 are orthogonal

An nxn symmetric matrix with real entries is always diagonalizable over the complex numbers in an ____ basis.

orthonormal basis

Given a symmetric matrix (A=A^T), we find an orthonormal matrix P and a diagonal matrix D both with real entries such that:

What is thefFirst step to diagonalization of symmetric matrices?

Show that all eigenvalues are real

Positive definite

symmetric matrix whose eigenvalues are all positive

Are all positive definite matrices invertible?

Yes det = +

Given any invertible matrix, ___ is positive definite

Cholesky decomposition

For any nxn symmetric matrix A, the following conditions are equivalent (and therefore any imply that the matrix is positive definite):

Cholesky Decomposition

Matrix representation of a quadratic function

f(x) = x^T A x

Given a quadratic function with f(x) = x^T A x where A is symmtric. The origin is a local min if?

A is positive definite

Given a quadratic function with f(x) = x^T A x where A is symmtric. The origin is a local max if?

–A is positive definite

Given a quadratic function f(x) = x^T A x and a circle ||x||=1, what is the max/min value of f?

Max = largest eigenvalue of A Min = smallest eigenvalue of A