3a midterm 2 part a

matrix

an array of numbers that represent a linear transformation that rotates, stretches, or shears vectors in space

identity matrix (I)

the matrix representation of the scalar 1(array of 1s on the diagonal and 0s everywhere else), multiplying any matrix by I gives you the same matrix. it is important when finding inverses because it is the target(what you’re trying to represent P as to get P^-1) and it appears in eigenvalue equations.

transpose (A^T)

transpose means to switch; a matrix operation that flips the matrix along its diagonal, making rows columns and columns rows

what is (A + B)ᵀ, and how do you solve it

A^T + B^T, you can either find add matrix a and b and transpose the sum, or transpose each matrix individually and find the sum

what is (AB)ᵀ

B^T*A^T

Does order matter in matrix multiplication? Meaning is AB=BA

yes, order matters because in matrix multiplication you’re usually applying the matrix that comes first to the matrix that comes second, and if you switch the order, there may not be enough numbers in columns or rows in a matrix to satisfy the composition of the other (rows=columns rule)

how do you recognize a diagonal matrix?

when you see a square matrix (same # of rows to columns) with all elements outside the main diagonal being 0

how do you multiply a matrix by a diagonal matrix?

if the diagonal matrix is on the left, meaning d*some matrix, you scale the rows. if the diagonal matrix is on the right, meaning some matrix*d, you scale the columns

matrix multiplication rule

number of columns in A = number of rows in B; the result is rows of A * columns of B

how to check if a matrix is invertible

determinant is a non-zero number

how would you find a matrix b given the matrix ab and matrix a

given A*B=AB, we have to isolate B. to do this, we must divide both sides by A(or multiply by the inverse). since you can’t divide matrices, we multiply both sides of the operation by the inverse of A, so we get B=A^-1*AB. to do this, we need to check if A^-1 exists so we calculate the determinant to confirm its a non-zero number. then, we calculate the inverse using the formula and multiply it by AB to find B

how do you find the inverse of a matrix

determine if it is square and has a non-zero determinant, then plug into 1/(ab-bc) * ([d -b][-c a])

how do you find the determinant

det(A)=ad-bc, given that matrix A is square

what is the inverse matrix method

method to solving systems of linear equations AX=B by finding A^-1 to solve X=A^-1B, applicable when the inverse exists (det not equal to 0)

determinant

factor that scales a matrix, either stretching or shrinking it. in a 2 dimensional view, it is the area of the parallelogram formed by column vectors

cofactor expansion

used to break down a 3×3 determinant into 3 2×2 determinants. you select the row or column that has the most zeros, say row a b c, and you multiply it against its minor, which is the 2×2 grid left behind when you remove the row and column the number is in. you then add or subtract these products based off the alternating sign matrix, [+ - +] [ - + -] [+ - +]

diagonal matrix, det?

all entries apart from the main diagonal must be 0; det=product of diagonal entries

triangular matrix, det?

either all entries above or all entries below the main diagonal must be 0(either is fine); det=product of diagonal entries

det of cA, given A is some matrix

c^n * det(A) for an nxn matrix

cramer’s rule, ie you’re given 2x₁ + x₂ = 5 and x₁ + 3x₂ = 7

solving a system of equations using determinants instead of row reduction. given 2 equations, separate the expression into matrix a and the value of the expression as matrix b. find the determinant of a. then to find x1, replace column 1 of a with b, and find its determinant. to calculate x1, divide that determinant by the determinant of A (like det A1/det A). repeat that same process for x2, replace column 2 of A with b and find its determinant, and then divide that value by the determinant of A.

eigenvalue λ

factor by which an eigenvector is scaled, satisfying the equation Ax = λx

what is the difference between a determinant and eigenvalues? consider the difference in terms of quantity and geometric interpretation

an eigenvalue is a scalar that determines by how much an eigenvector is scaled, satisfying the eigenvalue equation. a determinant is a single scalar value that tells you how area has been changed. an mxn matrix has n eigenvalues but only one determinant, and eigenvalues describe stretching/compressing along eigenvectors but determinants describe the total change in space.

how do you find det of a matrix using its eigenvalues?

det=product of eigenvalues

what is the relationship in invertibility between eigenvalues and determinants

if either the eigenvalue or determinant is 0, the matrix is non-invertible

eigenvector

the special non-zero vector v that is scaled when A acts on it; doesn’t change direction when having a linear transformation applied to it

does the presence of a zero eigenvector indicate non-invertibility

eigenvectors can never be 0 by definition, but if a zero eigenvector does indicate non-invertibility

how to find eigenvalues

1) det(A - (eigenvalue * I)=0

2) expand the determinant to get the characteristic polynomial

3) solve for the roots of the polynomial to get the eigenvalues

how to find eigenvectors after finding eigenvalues

for each eigenvalue, plug back into (A - (eigenvalue*I))v=0 and solve by row reduction. the free variables are your eigenvectors; you determine how many eigenvectors an eigenvalue has by counting the number of free variables that occur when you row-reduce.

when are eigenvalues linearly independent

eigenvectors corresponding to different eigenvalues are always linearly independent; if you have repeated eigenvalues you could get multiple eigenvectors for the same eigenvalue, so they may not be linearly independent.

how to determine if vectors are linearly dependent?

see if det = 0

how to check if an eigenvalue is an eigenvalue?

see if det(A - (eigenvalue * I)=0

when are matrices determined diagonalizable when considering eigenvalues and eigenvectors

a matrix is diagonalizable when you have enough independent eigenvectors for the given eigenvalues, including repeated ones (ie an eigenvalue with a multiplicity of 2 needs 2 independent eigenvectors, 2 eigenvalues need 2 independent eigenvectors)

diagonalization

finding a new coordinate system(set of basis vectors) to represent a matrix as a diagonal matrix with eigenvalues on the diagonal

when is a matrix diagonizable

iff you can find n linearly independent eigenvectors for an nxn matrix, for each eigenvalue its algebraic multiplicity must equal its geometric multiplicity, or the eigenvalues must be distinct(different)

what is an eigenvalue’s algebraic multiplicity

how many times it occurs in the root of the characteristic polynomial

what is an eigenvalue’s geometric multiplicity

the number of linearly independent eigenvectors that exist for a specific eigenvalue (the number of free variables)

change of basis for matrices

A squiggle = P^-1 * AP where P is the new basis vectors as columns and converts from new basis to standard and P^-1 converts from standard to new basis. the two matrices represented by P and P^-1 are similar and represent the same transformation in terms of different basis vectors.

what is diagonalization

type of change of basis, using the A=PDP^-1 formula to find a new set of basis vectors(new coordinate system) in which the matrix looks simpler with eigenvalues on the diagonal

when can you diagonalize a matrix

if you can find n linearly independent eigenvectors for an nxn matrix

what is the diagonalization process?

find the eigenvalues, then the eigenvectors. build the P matrix using the eigenvectors as columns, then D matrix using the eigenvalues on the diagonal. then find the inverse of P, p^-1 and verify that A=PDP^-1

what is a vector subspace (and its conditions)

subset of a vector space that is itself a vector space, closed under scalar addition and multiplication, and contains the zero vector. adding any two things in w stays in w, and scaling anything in w stays in w.

how do you check if something is a vector subspace; which condition is checked first

• it has the zero vector (plug in 0 for all the parameters and see if you get 0, you usually will unless you’re adding by a constant)

• it is closed under addition (if vectors u and v are in w, u+v must be within w)

• it is closed under multiplication (if a constant c is multiplied to an element in w v, cv must be in w)

• zero vector is checked first

how are subspaces and subsets related?

every subspace is a subset, but every subset is not a subspace. a subset is some collection of elements from V, a vector space. a subspace has to satisfy a set of 3 conditions, so a subspace is considered a special type of subset of a vector space.

in what case would you not perform computations to determine if something is a subspace

if its span

what is a linear map

a linear transformation; a function between two vector spaces that abides by vector addition and scalar multiplication.

how do you determine if something is a linear map

1) addition: multiplying a sum by T should give you the same result as multiplying each addend by T and then summing

2) scaling: multiplying a vector in by a constant and then mapping it to a matrix should give you the same final matrix as mapping the vector to it and then scaling by some constant

what is the geometric interpretation of linear maps

transformations of space that keep the origin fixed, and lines straight and parallel like rotations, reflection, scaling, or shears

is rref unique?

yes, rref is unique because irrespective of what transformations you perform on the matrix, you will always end up with the same results of 1s and 0s. ref is not unique because different operations can give you different refs.

what is one-to-one

different inputs give different outputs and no outputs are the same(no vectors get sent to the same place)

what is the kernel of a linear map

set of all vectors x that evaluate to 0, always a subspace containing the zero vector. it measures what occurs when the transformation collapses to 0, and if the kernel only includes the 0 vectory T is one-to-one

what is the geometric interpretation of kernel

if 3d space is being projected down to 2d space, meaning you drop the last coordinate, [0,0,z] becomes [0,0] and the x-axis lies in the kernel. this is why kernel is referred to as null space, it holds all the information that becomes lost.

how do you find the kernel given a matrix A

write out matrix A, row reduce to A | 0(set b column to 0), identify free variables, and write the general solution using s and t(these vectors span the kernel), separate the s and t pairs and write the kernel vectors

dimension of kernel rank nullity theorem

dim(kernel)+dim(column space) = number of columns in A

how would you check that 2 vectors constitute a basis for R², given they’re already in the dimension?

basis means linearly independent and span, given that 2 vectors are already in R², you would need to compute the determinant of the two bases and make sure its not 0 by putting the vectors as columns to determine if they’re linearly independent

how do you expand vectors in a basis?

expanding vectors in a basis means every vector can be written as some combination of basis vectors, you just need to find the weights c1 and c2 such that c1b1+c2b2=V. to do this, you set up a matrix [c1 c2| v] and row reduce to 0s and 1s to find the coordinates(aka find the coordinate vector), writing c1 and c2 as columns.

how do you find a change of coordinates matrix from basis B to basis C, given B was [1 -3][2 -5] and C was [1 -2][5 -6]?

set up [C | B] and reduce to [I | P] where c vectors are the left column and b vectors are the right. do rref to [I |P] where whatever P is, that is your final answer.

what is the difference between an upper triangular, lower triangular, and diagonal matrix

upper triangular matrix = all entries below the main diagonal as 0, lower triangular = all entries above the main diagonal are 0, and diagonal matrix means all entries apart from diagonal must be 0.

how do you find the determinant of a triangular or diagonizable matrix

product of the diagonal entries (even for 3×3 matrix)

how do you find the inverse of a diagonal matrix, what is key about diagonal matrices

replace each diagonal entry by its reciprocal; it acts on each entry individually

how do you find the eigenvalues of a diagonal matrix? why are they those values?

they’re the main diagonal; for triangular matrices, the eigenvalues are the diagonal entries

what would you do if you had to make a diagonal matrix non-invertible

its non-invertable when det(A)=0 so i would make one of the diagonal entries 0

what is a block diagonal matrix

a diagonal matrix but you have smaller matrices(blocks) on the diagonal instead of individual entries, and everything outside these smaller matrices are 0.

how do you find the det of a block diagonal matrix A? when is it invertible?

det(A) = det(block 1) + det(block 2); its invertible if each block is invertible(each det isn’t 0), and non-invertible otherwise

how do you find the inverse of a block diagonal matrix

find the inverse of each smaller matrix (2×2 formula) separately and put them into [block 1 inverse 0] [0 block 2 inverse]

how would you solve the equation ax=b if you were given a matrix for a and a vector for b

to solve for x, perform back substitution on each of the original blocks, mapping each one to a vector value in b(or you could find the inverse of a and multiply by b)