Linear Algebra Final Exam Review

vector

A quantity that has magnitude and direction

dimension of a vector

Number of components in a vector.

vector space

A collection of elements that can be formed by adding or multiplying vectors together.

dimension of a matrix

The number of rows and columns of a matrix, written in the form rows×columns.

Vector Addition

adding or combining quantities that have magnitude and direction

Vector Multiplication

not commutative

a x b =/= b x a

Linear Combination

A sum of scalar multiples of vectors. The scalars are called the weights.

span of vectors

Set of all linear combinations of vectors.

dot product

vector multiply vector

results in scalar quantity

= a b cos(angle)

unit vector

v/||v||

norm of a unit vector

sqrt(u*u)

distance between two vectors

d(u,v) = ||u-v||

orthogonal vectors

two vectors whose dot product equals zero

projection of u onto v

(u*v/|v|^2)v

orth v U

U - projection of u onto v

Diagonal Matrix

a square matrix whose entries not on the main diagonal are all zero

identity matrix

a square matrix that, when multiplied by another matrix, equals that same matrix

transpose of a matrix

Switch the rows and columns - imagine it kinda swinging up/down a 90 degree angle

dimension of A*B (matrix multiplication)

the row from A and the column from B

a system of equations can have

one unique solution

no solution

infinite solutions

homogenous system

always has zero vector as a solution

rref

Same as REF but +...

4. The pivot in each non-zero row = 1

5. Each pivot is the only non-zero entry in its column

REF of matrix

1. any row containing entirely zeros are at bottom

2. each leading entry of a row is in a column to the right of the leading entry of the row above it

Inverse Matrix

Matrix that, when multiplied, yields the identity matrix.

A is not invertible if...

1. A is not a square matrix

2. A is a zero matrix

3. A has a zero row or zero column

Determinant of a matrix

ad/bc

Steps to take an inverse of a matrix

use the Gauss-Jordan Elimination method by augmenting your matrix A with the identity matrix [A|I] and performing row operations to transform it into [I|A⁻¹]

finding inverse of a 2x2 matrix

[[a, b], [c, d]]

Determinant: |A| = (ad) - (bc).

Adjoint: Swap a and d, then negate b and c: [[d, -b], [-c, a]].

Inverse: A⁻¹ = (1/|A|) * [[d, -b], [-c, a]]

determinent and row operations: switching two rows

multiples determinent by -1

determinent and row operations: multiplying by a scalar

multiples determinent by the scalar

determinent and row operations: adding a multiple of one row to another

does not change determinant

det(AT)

det(A)

det(A^-1)

Det(A)^-1

determinant of upper/lower triangular matrix or diagonal matrix

the product of all diagonal enteries

cofactor expansion steps:

select any row or column of a matrix, multiply each element in that row/column by its corresponding cofactor, and then sum these products

transformation of a matrix

R^k -> R^n

also called mapping

domain

Set of all possible input values.

codomain

the set of all possible outputs for a function

Range(Ta) or Image(Ta)

set of actual outputs

Range(Ta) for linear systems

vectors for which system Ax=b is consistent

Range(Ta) for linear combinations

If TA is associated to A, then the range(TA) is the span of the columns of A, i.e., the set of all linear combinations of the columns of A.

is b in the range of Ta?

asking if there is a v such that Ta(V) = b

(so augment b to A and then solve)

if there is a solution, it is in range

linear transformation: rotation

R(θ)=[cos⁡θ −sin⁡θ sin⁡θ cos⁡θ]

Clockwise rotation: just flip the sign of θ

reflection across x axis

[1 0 0 −1]

reflection through origin

[-1 0 0 -1]

reflection across y axis

[−1 0 0 1]

reflection across the line y = x

[0 1 1 0]

scaling

[sx 0 0 sy]

sx scales horizontally

sy scales vertically

horizontal shear

[1 k 0 1]

vertical shear

[1 0 k 1]

projection onto x axis

[1 0 0 0]

projection onto y axis

[0 0 0 1]

translation

[1 0 tx 0 1 ty 0 0 1]

txt_x = how far you move a point in the x‑direction (horizontal shift).

tyt_y = how far you move a point in the y‑direction (vertical shift).

properties of linear transformations

Additivity: T(u+v)=T(u)+T(v)T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) → The transformation distributes over vector addition.

Homogeneity (scalar multiplication): T(c⋅u)=c⋅T(u)T(c \cdot \mathbf{u}) = c \cdot T(\mathbf{u}) → Scaling before or after the transformation gives the same result.

Preserves the origin: T(0)=0T(\mathbf{0}) = \mathbf{0}. → Linear transformations always map the zero vector to itself.

Matrix representation: Every linear transformation T:Rn→RmT: \mathbb{R}^n \to \mathbb{R}^m can be written as T(x)=AxT(\mathbf{x}) = A\mathbf{x}, where AA is an m×nm \times n matrix.

Composition corresponds to matrix multiplication: If T1(x)=AxT_1(\mathbf{x}) = A\mathbf{x} and T2(x)=BxT_2(\mathbf{x}) = B\mathbf{x}, then (T2∘T1)(x)=B(Ax)=(BA)x(T_2 \circ T_1)(\mathbf{x}) = B(A\mathbf{x}) = (BA)\mathbf{x}.

Invertibility: A linear transformation is invertible if and only if its matrix is invertible (determinant ≠ 0). → Inverse transformation corresponds to the inverse matrix.

Determinant meaning:

∣det⁡(A)∣|\det(A)| = scaling factor of area/volume.

det⁡(A)<0\det(A) < 0 → orientation is flipped (reflection).

det⁡(A)=0\det(A) = 0 → transformation squashes space into lower dimension (not invertible).

kernel(T) or null space of A

vectors mapped to 0 by T

so Av=0

a transformation is onto if

Av = b has at least one solution

one-to-one

Av = b should have at most one solution

vector subspace

collection or set 𝑈 ofvectors in 𝑉, such that• adding two vectors in 𝑈 yields a vector in 𝑈• scaling a vector in 𝑈 yields a vector in 𝑈

basis

linearly independent spanning set

linearly independent

the only solution to

c1v1+c2v2+⋯+cnvn=0

is when all coefficients are zero:

c1=c2=⋯=cn=0

Square matrix → check determinant.

Non‑square matrix → check rank (full column rank = independent).

Always → solve Ax=0 if unsure. (rref should show that each x variable is 0)

dimension of a subspace

number of vectors in that basis

let v be a subspace of R^n and dim(V) = m

basis must have exactly m vectors

spanning set must have at least m vectors

linearly independent set must have at most m vectors

column space

Put the matrix AA into row‑reduced echelon form (RREF).

Identify the pivot columns (the columns that contain leading 1's).

The corresponding columns in the original matrix (not the reduced one!) form a basis for the column space.

Dimension: The number of pivot columns = rank of A

row space

Row reduce AA to RREF.

The non‑zero rows of the RREF form a basis for the row space.

Dimension: The number of non‑zero rows = rank of A

rank(A) =

dim(col(a)) = dim(row(a)) = dim(col(AT)) = rank(AT)

rank-nullity theorem

rank(A) + nullity(A) = number of columns, where rank is number of pivot columns, and nullity is number of free variables

change of basis

Form the change of basis matrix

Put the new basis vectors as columns in a matrix P. Example:

P=[1 1 1 −1]

Convert coordinates from new basis to old basis

If a vector has coordinates [v]B in the new basis, then in the standard basis:

[v]standard=P[v]B[v]

Convert coordinates from old basis to new basis

Use the inverse:

[v]B=P−1[v]standard

Gram-Schmidt Process

Start with a set of linearly independent vectors {v1,v2,...,vn}

Define u1=v1

u2 = v2 - proju1V2

u3 = v3 - proju1V3 - proju2V3

then {u1, u2, ..., un} is an orthogonal basis for v

Least squares

Set up normal equations: ATAx=ATb

Solve for x

Equivalent to projecting b onto the column space of A

inner products

⟨u,v⟩=uTv

properties of inner products

Symmetry: ⟨u,v⟩=⟨v,u⟩

Linearity: ⟨au+bv,w⟩=a⟨u,w⟩+b⟨v,w⟩

Positive definiteness: ⟨v,v⟩≥0, equality only if v=0

Defines length: ∥v∥=sqrt(⟨v,v⟩)

Defines orthogonality: ⟨u,v⟩=0

distance = sqrt(

eigen problems

Solve det⁡(A−λI)=0 for eigenvalues λ

For each λ, solve (A−λI)v=0 for eigenvectors.

eigenvalues of triangular matrix

just the enteries on the diagonal

product of eigenvalues

determinent of A

diagonalisation

Find eigenvalues and eigenvectors.

Form matrix P with eigenvectors as columns.

Then A=PDP−1, where D is diagonal with eigenvalues.

Markov Process

Transition matrix P with nonnegative entries, columns (or rows) sum to 1.

State vector evolves as xk+1=Pxk

Long‑term behavior: look at steady state x such that Px=x

SVD

Factor A into A=UΣVT

U: orthogonal matrix (columns = left singular vectors).

Σ: diagonal with singular values. (sqrt of eigenvalues of ATA)

V: orthogonal matrix (columns = right singular vectors).

QR factorization

Apply Gram-Schmidt to columns of A.

Get orthonormal matrix Q.

Upper triangular matrix R satisfies A=QR

transition matrix P from B2 to B1

P = B1^-1 * B2

Guaranteed eigenvector of A^n

it's just the original eigenvector x

If λ is an eigenvalue of A

λ^k is an eigenvalue of A^k

Suppose A is mxn, B is nxp, and C is pxq

then ABC is mxq

if v and w are parallel, then projvW = ?

w

(AB)^T

B^TA^T

If A*B = 0 and A is invertible

then B is 0

det(2A)

2^n(det(A))

det(A^-1)

1/det(A)

What is the angle between a vector ⃗v and its unit vector ⃗v/ |⃗v |

Zero, since they are in the same direction