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CAL / Math / MATH 54 / How do you prove a vector space over a field?

How do you prove a vector space over a field?

How do you prove a vector space over a field?

Description

School: University of California Berkeley
Department: Math
Course: Linear Algebra and Differential Equations
Professor: Vivek shende
Term: Winter 2016
Tags: Linear Algebra
Cost: 50
Name: Midterm 2 Study Guide - Math 54
Description: A study guide covering topics in midterm 2 for Lin Lin's Math 54. Bases, Vector spaces, Transformations, Change of Basis, Determinant, Eigenvalues and Eigenvectors, Diagonalization, Geometry.
Uploaded: 10/19/2017
14 Pages 38 Views 4 Unlocks
Reviews


Midterm 2 Study Guide Monday, October 16, 2017 9:52 PM


How do you prove a vector space over a field?



A.

Vector Spaces

a.

A vector space is closed under addition, linear combination, scaling, and contains the  zero vector

B.

Basis for vector spaces

a. A set of vectors that are linearly independent that span a vector space.

b.

A standard basis for a normal Rn*n vector space is made up of vectors: e1, e2, …, en.

i. Where:

We represent a number in a basis in matrix form as a collection of values for each  

c.

basis vector.

What you'll be asked:

i. Find [x]B, or the vector in the standard basis for P2, representing this polynomial


How do you find the basis of a vector space?



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ii.

Find the matrix for the transformation that represents going from a third degree  polynomial to its derivative.

We know that all of the first term of P3 has derivative 0.

The second term maps to the first term of P2, and is multiplied by 1 (exponent 1) The third maps to the second, and is multiplied by 2.

The fourth maps to the third, and is multiplied by 3.

Is 1 + x ­ x3 in the span of {1+x3, 1­2x, x2­x3 iii. }?

iv. Are the following matrices linearly independent?

C.

Definitions for bases

a. The dimension of a vector space is the number of basis vectors that it has. The image of a transformation is its range (for example, R3 b. , or P2)


How do you prove a set of vectors is a basis?



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c.

The null space:

i. Of a transformation is called the kernel

ii.

Is spanned by a set of vectors (nonzero) that, under the transformation, yield  the zero vector

iii. Is also a subspace, so it can have a dimension

d.

For the transformation in B. c. ii. above, what is the null space, image, and dimension  for both the null space and the input?

D.

Transformations terminology

a.

An isomorphism is simply a transformation/matrix that behaves the same way. It must have the same dimension. Rigorously:

i.

Def. A linear transformation T: V ­> W is called an isomorphism if there is a  linear transformation called T­1:W ­> V such that:

T(T­1(w)) = w for any w in W.

T­1(T(v)) = v for any v in V.

b.

Note that if a transformation doesn’t have an inverse, then it doesn't have an  isomorphism.

E.

Matrix representation of Linear transformations

a.

To do a transformation from one basis to another, where the transformation T is in a  standard basis: Don't forget about the age old question of What are the three components of the quebec legal system?

b. The dim(V) = dim(null V) + dim(rank V)

You will be asked:

c.

i. Given a basis, what is its corresponding transformation ii. Solve the below: Don't forget about the age old question of What is damköhler number?

F.

Determinant

a.

The determinant is an intrinsic property of matrices that represents how  an nxn matrix (as a transformation) scales the parallelopiped formed by  the column vectors. However, it is the signed area formed by these  vectors, which means it can be negative. If you want to learn more check out List of countries that belongs to south asia?

b.

Calculating the determinant.

For 3x3 matrices and larger

i.

First, choose a column or row to expand upon (you want the  greatest number of zeroes in it)

ii.

For each number along that column or row, multiply that number  by the determinant of the rest of the matrix (you delete the column  and row of each number per number)

iii.

Continue until you've completed your column/row, adding or  subtracting each value according to alternating +/­ in a matrix (note  that lots of zeros are useful here because they will add zero to your  total sum)

c. If the determinant is nonzero, then the matrix is invertible. d. det(AB) = det(A)det(B) (this is not true for adding)

e.

Determinant of an upper or lower triangular matrix is the product of the  diagonal down the matrix.

f.

For a matrix, the effect on the determinant:

i. Add scaled row to another (replacement): no effect

ii. Swap rows (interchange): det > ­det

iii. Scale row: det > scaling factor * det

Det(kA) = Det(A)* kn g. , where A is nxn

Cofactor matrix is the matrix made up of, for each location i, j in A, the  

h.

determinant of Aij.

i. The transpose of the cofactor matrix of A is the adjugate of A A­1 j. = (1/det(A))*adj(A) ­­­ variation of cramer's rule k. Cramer's rule:

G.

Eigenvalues and Eigenvectors

a.

For any nxn matrix, you can treat it like a transformation, which changes  space; some of these stretch certain vectors in their domain. b.

For some of these matrices/transformations, there are vectors that can  be input which will result in an output that is simply a multiple of the  input vector. This vector is an eigenvector.  

i.

If one eigenvector exists, by scaling, so do every multiple of the  eigenvector. We usually write down the one with all whole  numbers.

c.

If the eigenvectors of a matrix span the vector space of the matrix, then  the matrix is diagonalizable

d.

Each eigenvector is scaled by a certain amount by the matrix  transformation. This amount is called an eigenvalue.

H.

Calculating Eigenvectors/Values

a.

The multiplicity of this characteristic polynomial:

i.

1<= geometric multiplicity (number of unique roots) < algebraic multiplicity (total number of roots counting repeats such as (x­2)2) ii.

*note characteristic equation:

det(A­λI) = 0

b. The eigenvalues of a triangular matrix are the values on the diagonal.

c.

If there are r distinct eigenvalues, there are at least r distinct  eigenvectors. The set of these r corresponding eigenvectors are linearly  independent.

Recall determinants:

d.

Det(A) = (each scaling factor multiplied together)*(­1)number of times row i.

swapped*(product of pivots in U)

1) Where U is an echelon form of A 2) Where A is invertible

e.

A and B are similar if there exists A = PBP­1.  

i.

If A and B are similar, they have the same characteristic polynomial  and eigenvalues with same multiplicities, but not the other way  around

Guaranteed diagonal matrix, assuming A is diagonalizable: I.

A = PDP­1.

a.

i. D is made up of the eigenvalues arranged along the diagonal ii. P is made up of each corresponding eigenvector in order iii. (the above refer to the eigen­ of A)

J. Properties and methods

K. Dot product

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