Linear Algbera 1411A

What is the direction of flow through batteries

negative to positive

What is transposition of A and its properties

• dim(A) = m x n → dim(A^T) = n x m

• (A^T)^T = A

• column 1 becomes row 1

What is an identity matrix

• equivelent to 1 in matrix language

• 1 across diaganol and zero everywhere else

• n x n matrix

To be an invertible matrix? and what they can be used for?

• matrix must be square

• determinant must be non-zero

• there cannot be a row or column of zeros

• A^-1 (A) = I_n

• A (A^-1) = I_n

• Use to solve linear systems (for x bar) when Ax = b, by multiplying both sides by inverse A^-1 (A)x = A^-1 (b) → x = A^-1 b

• Can also be used to find elementary matrix since E(A) = B

Properties of elementary matrices

• E = I_n with one elementry row operation

• All E are invertible.

Properties of Invertible matrices A and B

(A x B)^-1 = B^-1 x A^-1

If A^T is invertible too that (A^T)^-1 = (A^-1)^T

What are diagonal matricies and what are its properties?

square matrices w/ its only non zero elements on the main diagonal

A = diag (a1, a2… an)

• A^k = diag(a1^k, a2^k… an^k)

• Diagonal matrices are invertible if all main diagonal elements are nonzero

  • A^-1 = diag(1/a1, 1/a2, … 1/an)

What are the triangular matricies

n x n

Upper Triangle = all elements below main diagonal are zero

Lower Triangle = all elements above main diagonal are zero

Properties of Triangular Matricies

• Transpose of UTM is a LTM and vice versa

• Multyplying two LTM matrices results in a LTM matrix

• Multiplying two UTM matrices results in a UTM matrix

• Traingular Matrix is invertible if all main diagonal elements are nonzero (use regular inverse block method to find.

  • A^-1 will be same type of triangular matrix

Syemmetric matrices and its properties

A is symmetric is A^T = A. There is a mirror over main diagonal.

• For A and B being Sysmteric nxn:

  • A +-B, constant(A), A^-1 are also symmetric.

  • A x B not always symmetric

• For any matrix B

  • B x B^T, B^T x B is symmetric

Free vars

• variables without leading one

• if a matrix has a free var than it has infinate solutions

Homogenous metrices

all b’s = 0

System is consistent (solution of vars = 0)

There are nontrivial solutuons (sols other than 0) if the # vars > # equations

What is a determinant

measures how much a transformation scales an area (2D) or volume (3D)

Properties of determinants

det( A x B) = det(A) x det(B)

det(A^T) = det(A)

det(cA) = cⁿ det(A)

What are the properties of determinants when elementary operations are applied?

• interchange two rows → multiply det by (-1)

• multyply row by constant c → multiply det by (c)

• add multiple of one row to another → det remains unchanged

What is Rⁿ

Rⁿ is a vector space formed by n-tupples (x bar) - a collection of all possible var value columns with n real numbers

What are standard basis vectors

e hat 2 = (0, 1, 0, 0 … 0) → two corresponds to position of 1

These are the building blocks of x bar . A vector x is some amount of each building block.

x bar = x1 e1 + x2 e2…+ xn en

What is function transformation

functions f is a “rule” that transforms Rn → Rm

• Rn is domain, Rm is codomian/range

Use matrix multiplication as a rule T_a(x) = A(x) = result b, with b in Rm.

• When m =/ n → A transforms points from Rn to Rm (the m dimension of A)

• When A = 0 mxn, any x inputs = 0 output

• When m = n, the result is in Rn still

• When A = In, result is still x

What are equivalent vectors?

the magnitude and directoin of X and Y are same but has a different starting point in space (slides around)

Algebraic properties of vectors

Addition

• Cummatative

• Associative

Scaling

• length gets multipled by constant

What is linear combination

if you have a set of vectors in Rn = xbar 1, x bar 2…

and a set of constants: c1 c2…

scale each vector by matching constant then add them up

c1 xbar1 + c2 xbar2…

What is the distance and norm

Distance is how far apart two points are and it depends on space Rn

Norm of a vector is its length (disregarding direction)

What is the unit vector

when the norm ||x|| = 1

for example basis vectors.

What is normalizing vectors

take a nonzero vector and srhink/stretch it to norm = 1 without changing direction

• for x bar to have a norm of one = x bar / ||x|| → vector divided by norm

What is the triangular inequality

For any two vectors the direct path is shorter: ||u|| + ||v|| >= ||u+v||

What is the formula for the dot product

Geometrically: a dot b = ||a|| ||b|| cos(theta) ← theta is the angle between the vectors

Algebriacally (with columns): a dot b = a1b1 + a2b2 +… +anbn

Answer is a number not vector.

What does the dot product do?

1. Test for perpendicularity. if dot product = 0 → cos(theta) = 0 meaning that a is perpendicular to b → which means they are orthogonal.

2. Find angle between vectors by solving for cos(theta) by putting the geometric and algebraic definations equal to each other.

What is orthogonality

Two vectors are perpendicular.

nonzero vectors u dot v = 0 are orthogonal

used to define planes/points/lines in space.

Example- standard basis vectors when multiplied are either 0 (perpendicular) or 1 (actually the same vector)

What is the equation of a plane

use a vector perpendicular to plane (check using dot product = 0) v and a point x

v1x1+ v2x2 + v3x3 = d, where d is the dot product of v dot x

What is the projection theorem

finding the length of w1 the projection of w onton u = proj_u w

= u(u dot w)/ ||u||²

What is the general vector space

vectors are just an element of vector space. the vector is an example of the objects of the space (the choosen collection or rule)

What are the rules to see if V is a vector space

u, w, v are elements of V, c is constant

1. u + w = w + u order doesnt matter

2. u + (w + v) = (u + w) + v grouping doesnt matter

3. there is a zero element u + 0 = u

4. for each element of V there is the opposite elemetn u + (-u) = 0

5. (c+c1)u = cu +c1u distributive

6. (c x c1)u = c(c1 x u) order doesnt matter

7. 1 x u = u

8. c(u + w) = cu + cw

9. when adding two elements of V the result is still within V

10. when multiplying to elements of V the result is still within V

What are the common vector spaces?

• Rn R2 R3

• zero vector space (single element 0)

• Matrix spaces

• Single variable functions

What is a subset vs subspace

subset is any collection of elements drawn from V

a subset is a subspace if that subset is a vector space in its own right.

What are the conditions for a subset W of V to be a subspace?

1. W is inside of V

2. closed addition (two elements of W added together are still within W)

3. closed multiplication (two elements of W multiplied are still within W

4. contains a zero element (if a line doesnt go through the origin its not a supspace)

Subspaces of R2

• zero

• straight line through origin

• R2 itself

Other examples of subspaces

Within the vector space of matrices → symmetric matricies

Within functions → continous functions

Within functions → within continous functions → polynomial functions

Intersection of subspaces → the elements within two subspaces (overlapping venn diagram) create a new subspace.

What is span

The span of a set of vectors is the set of all possible linear combinations of those vectors → which forms a subspace W = span(S) ← S is a spanning set for W

W = c + c1s1 + c2s2… cnsn

S = 1, s1, s2, … sn ← all different (no 2s2 or 0)

How to solve for linear combinations

set known vectors as the coeff. for the constant c’s ans sols as b

v(c ) = b

if there is a consistency statement but has one failure it is not a span