Courses › MATH 230

About this course

This is a free self-study course outline written by Saving Planets. It is not a university course, it carries no academic credit, and no school is involved.

Its scope follows a one-semester sophomore linear algebra course. It is modeled on the topics and depth of MIT 18.06 Linear Algebra taught by Gilbert Strang (free on MIT OpenCourseWare), and on the Georgia Tech and Purdue sophomore linear algebra courses, using the free Georgia Tech text Interactive Linear Algebra by Dan Margalit and Joseph Rabinoff and Jim Hefferon’s free Linear Algebra, which we link where it helps. None of those authors or schools endorses this page.

What we add: every unit ties to real planet work with a cited public source (GPS, Landsat 9, NASA sea-level satellites, NOAA weather supercomputers, EPA input–output and recycling data, USGS building studies, NASA’s EMIT mineral spectrometer), every number on the page is re-computed by a checking script before it goes live, and every lab runs in a free spreadsheet or on paper.

Work through it in order. Each unit has goals, key ideas, equations, step-by-step examples, practice with full answers, common mistakes, flashcards and a quiz. Your progress is saved only in this browser.

How this course helps save the planet

Linear algebra is the language of data about the planet. A satellite pixel is a vector, a GPS fix is the solution of a linear system, a weather forecast is millions of equations solved again and again, and an economy’s pollution can be traced with one matrix.

The same few ideas, elimination, projection, eigenvalues and the SVD, turn raw measurements into maps of forests, sea-level trends and mineral dust. Each unit ends with a short, sourced section. Values that are not from a source are labeled as sample values.

Before you start

Prerequisites

High-school algebra: solving two equations in two unknowns, and working with functions.
Calculus I helps with limits and derivatives in a few examples, but no calculus is needed for most of the course.
Comfort with a spreadsheet; Intro to Programming helps for the optional Python versions of the labs.
Calculus III and Differential Equations pair well with this course but are not required.

How to study

Do every computation by hand first on small (2 × 2 and 3 × 3) examples, then check it with a spreadsheet or calculator.
Think of Ax two ways: as dot products of the rows with x, and as a combination of the columns of A.
Check every answer: plug x back into Ax = b, multiply A by its inverse, or test Av = λv.
Keep track of sizes: an m × n matrix times an n × p matrix gives m × p, and sizes that do not match mean an error.
Draw pictures in 2D and 3D: lines, planes, projections and stretches make the rules easy to remember.
Do every practice problem on paper before opening the answer, and retake the quizzes a few days later.

Lab equipment (for the wet labs)

A computer with free spreadsheet software (its MMULT, MINVERSE, MDETERM and LINEST functions do matrix work).
Graph paper, a ruler, a protractor and a pencil.
Optional: free Python with NumPy for the larger versions of the labs.
A string and push pins or tape for the trilateration lab, and a printed map or sheet of paper.
A grayscale photo or a small grid of numbers for the image-compression lab.

Suggested 15-week schedule

About 6 to 9 hours a week: reading, practice problems and, in lab weeks, the lab.

Week	Unit	Topics	Reading
1	Vectors: Linear Combinations, Lengths and Dot Products	Vectors, linear combinations, span, length	MIT 18.06 L1; ILA 2.1–2.2
2	Vectors: Linear Combinations, Lengths and Dot Products	Dot products, angles, orthogonality; Lab A; start elimination	ILA 6.1
3	Systems of Linear Equations and Gaussian Elimination	Elimination, echelon forms, solution sets; Lab B	MIT 18.06 L2; ILA 1.1–1.3
4	Matrix Algebra: Multiplication, Inverses and LU Factorization	Matrix multiplication, inverses, A = LU; Lab C	MIT 18.06 L3–4; ILA 3.4–3.5
5	Linear Models: Input–Output (Leontief) Economics, Networks and Balancing	Input–output models, networks, balancing	Hefferon Topics; EPA USEEIO
6	Vector Spaces, Subspaces, Independence, Basis, Rank and the Four Fundamental Subspaces	Subspaces, column space, null space, Ax = 0	MIT 18.06 L5–7; ILA 2.6
7	Vector Spaces, Subspaces, Independence, Basis, Rank and the Four Fundamental Subspaces	Independence, basis, dimension, rank, four subspaces; midterm practice	MIT 18.06 L8–10; ILA 2.5, 2.7, 2.9
8	Orthogonality, Projections, Least Squares and Gram–Schmidt	Orthogonality and projections	MIT 18.06 L14–15; ILA 6.3
9	Orthogonality, Projections, Least Squares and Gram–Schmidt	Least squares, Gram–Schmidt, A = QR; Lab D	MIT 18.06 L16–17; ILA 6.4–6.5
10	Determinants: Properties, Cofactors, Cramer’s Rule and Volume	Determinants, cofactors, Cramer’s rule, volume	MIT 18.06 L18–20; ILA 4.1–4.3
11	Eigenvalues and Eigenvectors	Eigenvalues, eigenvectors, power iteration, modes	MIT 18.06 L21; ILA 5.1–5.2, 5.5
12	Diagonalization, Matrix Powers, Markov Chains and Linear Dynamical Systems	Diagonalization, Markov chains, dynamical systems; Lab E	MIT 18.06 L22–25; ILA 5.4, 5.6
13	The Singular Value Decomposition, Principal Component Analysis and Positive Definite Matrices	Positive definite matrices and the SVD	MIT 18.06 L27–29
14	The Singular Value Decomposition, Principal Component Analysis and Positive Definite Matrices	Low-rank approximation, PCA, condition number; Lab F	MIT 18.06 L29; Strang book site
15	Review	Review and final practice set	All units

Your progress

Tick units and labs as you finish them, and take the quizzes. Progress and quiz scores are saved only in this browser, never sent to us.

Turn on JavaScript to track progress.

The lab (Computation lab)

The lab is the hands-on part of the course. Here it is free and for practice only, with no credit.

Every lab runs in a free spreadsheet or on paper; Python versions are optional. Take regular breaks from the screen, and handle pins and scissors with care.

Safety reference: OSHA: Computer workstations eTool

Lab A: Pixel vectors and a vegetation index

Week 2

Goal: Treat satellite pixels as vectors: compute lengths, angles between pixels and the NDVI, and see that vegetation, soil and water point in different directions.

\cos\theta=\frac{\mathbf{p}\cdot\mathbf{q}}{\|\mathbf{p}\|\|\mathbf{q}\|},\quad \text{NDVI}=\frac{\text{NIR}-\text{red}}{\text{NIR}+\text{red}}

Angle between pixels and NDVI

Procedure

In a spreadsheet, enter three sample pixels as rows of (green, red, NIR) reflectances: vegetation (0.10, 0.08, 0.40), soil (0.20, 0.25, 0.30), water (0.06, 0.04, 0.02).
Compute each pixel’s length with SQRT(SUMSQ(…)) and each pair’s dot product with SUMPRODUCT.
Find the angle between each pair with ACOS and DEGREES.
Compute NDVI for each pixel. Optional: download a free Landsat scene and repeat for real pixels.

Data table

Pixel	Green	Red	NIR	NDVI
Vegetation	0.10	0.08	0.40	0.667
Soil	0.20	0.25	0.30	0.091
Water	0.06	0.04	0.02	−0.333

Sample reflectances chosen to look like typical vegetation, bare soil and water.

Sample calculation

Vegetation NDVI = (0.40 − 0.08)/(0.40 + 0.08) = 0.667.
Soil NDVI = 0.05/0.55 = 0.091; water NDVI = −0.02/0.06 = −0.333.
Angle vegetation–soil: dot = 0.160, lengths 0.420 and 0.4387, cos θ = 0.8683, θ = 29.7°.

Safety

Take a short screen break every half hour.

Home or virtual alternative

The whole lab runs in a free spreadsheet.

PhET: Vector Addition · USGS: Landsat 9

Planet link: USGS lists Landsat 9’s red (band 4) and near-infrared (band 5) bands, and its data are free, so anyone can map vegetation this way. USGS: Landsat 9

I completed this lab (wet or virtual)

Lab B: Trilateration, from strings to a linear system

Week 3

Goal: Find a position from distances, first with strings on a map as in the GPS.gov activity, then by turning the circle equations into a linear system and solving it by elimination.

2(a_2-a_1)x+2(b_2-b_1)y=d_1^2-d_2^2+a_2^2-a_1^2+b_2^2-b_1^2

Difference of two circle equations

Procedure

Print the GPS.gov trilateration map (without scaling) or mark three “satellites” on graph paper at (0, 0), (10, 0) and (0, 10) cm.
Convert signal times to distances with d = ct (c = 299,792,458 m/s); for the paper version use the sample distances below.
Draw circles with a string and pencil and mark where they cross.
Subtract the circle equations in pairs to get two linear equations, solve them by elimination in a spreadsheet, and compare with your drawing.

Data table

Beacon	Position (cm)	Distance (cm)	d²
1	(0, 0)	5.000	25
2	(10, 0)	8.062	65
3	(0, 10)	6.708	45

Sample distances for a receiver at (3, 4) cm. The GPS.gov activity uses four satellites on a map at 1:21,283,839.

Sample calculation

Eq. 1 − Eq. 2: 20x − 100 = −40, x = 3. Eq. 1 − Eq. 3: 20y − 100 = −20, y = 4.
GPS.gov time A = 0.00505783 s gives d = 1,516,299 m; at 1:21,283,839 that is 7.12 cm on the map.

Safety

Handle push pins and scissors with care.

Home or virtual alternative

The string-and-paper version is the home lab.

GPS.gov: Trilateration activity · Interactive Linear Algebra, 1.2 Row reduction

Planet link: GPS.gov designed this activity with NOAA; real receivers solve the same kind of system, with a clock unknown, many times a second. GPS.gov: Trilateration (NOAA classroom activity) · GPS.gov: Space segment

I completed this lab (wet or virtual)

Lab C: Matrices as transformations of the plane

Week 4

Goal: See matrices act on shapes: stretch, shear, rotate and reflect a square on graph paper, and confirm that areas scale by |det A|.

\text{area}(AS)=|\det A|\,\text{area}(S)

Area scale factor

Procedure

Draw the unit square with corners (0, 0), (1, 0), (1, 1), (0, 1).
Multiply each corner by A = [[2, 1], [0, 1]] (a shear and stretch), by R = [[0, −1], [1, 0]] (a 90° rotation) and by F = [[1, 0], [0, −1]] (a reflection).
Draw each image and find its area by counting squares or with the shoelace formula.
Compare each area with |det| and note which maps flip orientation (det < 0). Then check det(AR) = det A det R.

Data table

Matrix	det	Image area	Orientation
A = [[2, 1], [0, 1]]	2	2	kept
R (90° rotation)	1	1	kept
F (reflection)	−1	1	flipped
AR	2	2	kept

Exact values for the unit square.

Sample calculation

det A = 2 × 1 − 1 × 0 = 2; the image is a parallelogram with base 2 and height 1.
det(AR) = det A × det R = 2 × 1 = 2.

Safety

None beyond normal desk work.

Home or virtual alternative

Graph paper and a pencil are all you need.

Interactive Linear Algebra, 3.1 Matrix transformations · Interactive Linear Algebra, 4.3 Determinants and volumes

Planet link: Map projections and image registration use these same matrices; the determinant keeps area measurements of forests and floods honest. USGS: Landsat 9

I completed this lab (wet or virtual)

Lab D: Least-squares fit of sea-level data

Week 9

Goal: Fit a line, and then a parabola, to sea-level data with the normal equations, and compare with the spreadsheet’s built-in fit.

A^TA\hat{\mathbf{x}}=A^T\mathbf{b},\quad A=[\,\mathbf{1}\ \ \mathbf{t}\,]

Normal equations for y = C + Dt

Procedure

Enter the sample data below (or download NASA’s record from its Vital Signs page): t in years since 1993 and h in mm.
Build A with a column of ones and a column of t. Compute AᵀA and Aᵀb with MMULT and TRANSPOSE, then x̂ = MINVERSE(AᵀA)·Aᵀb.
Check against LINEST or a chart trendline. Plot the residuals b − Ax̂.
Add a t² column and refit. A positive t² coefficient means the rise is speeding up.

Data table

t (years since 1993)	0	8	16	24	32
h (mm)	0	22	48	78	114

Sample values shaped like the NASA record (which reports 114.4 mm of rise since 1993); use the real data for a real answer.

Sample calculation

t̄ = 16, h̄ = 52.4; Σ(t − t̄)(h − h̄) = 2272, Σ(t − t̄)² = 640.
D = 2272/640 = 3.55 mm/yr; C = 52.4 − 3.55 × 16 = −4.4 mm.
The residuals are positive at both ends and negative in the middle: a curve fits better, as NASA’s rising rate suggests.

Safety

Take a short screen break every half hour.

Home or virtual alternative

The whole lab runs in a free spreadsheet.

NASA: Global mean sea level · Interactive Linear Algebra, 6.5 Least squares

Planet link: NASA reports the rate of rise grew from 0.20 cm per year in 1993 to 0.44 cm per year now, an acceleration you can detect with a t² column. NASA: Global mean sea level (Vital Signs)

I completed this lab (wet or virtual)

Lab E: A Markov model of materials and recycling

Week 12

Goal: Build a transition matrix for material in use and lost to landfill or burning, use matrix powers to follow it over time, and see how a higher recycling rate keeps material in use longer.

\mathbf{x}_{k+1}=P\mathbf{x}_k,\quad P=\begin{bmatrix}1-f(1-r)&0\\f(1-r)&1\end{bmatrix}

Two-state chain: in use, lost

Procedure

Choose a sample discard rate f = 0.2 per year and a recovery rate r = 0.321 (EPA’s 2018 recycling-plus-composting share).
Build P in a spreadsheet and start with x₀ = (1, 0): all material in use.
Use MMULT to step forward 10 years, and compare with the formula (1 − f(1 − r))ᵏ for the share still in use.
Repeat with r = 0.5 and compare the average years in use, 1/(f(1 − r)).

Data table

Recovery rate r	Stay-in-use probability	In use after 10 years	Average years in use
0.321	0.8642	0.232	7.36
0.50	0.9000	0.349	10.0

Sample model: the discard rate f = 0.2 is a sample value; r = 0.321 is the EPA 2018 share.

Sample calculation

Stay probability = 1 − 0.2 × 0.679 = 0.8642; 0.8642¹⁰ = 0.232.
Average years in use = 1/0.1358 = 7.36; at r = 0.5 it is 1/0.1 = 10.0 years.
λ = 1 belongs to the “lost” state, which absorbs everything in the long run; λ = 0.8642 sets how fast.

Safety

Take a short screen break every half hour.

Home or virtual alternative

The whole lab runs in a free spreadsheet.

Interactive Linear Algebra, 5.6 Stochastic matrices · EPA: National overview of materials and waste

Planet link: EPA reports a 32.1% recycling and composting rate in 2018 and 50% landfilled; models like this show what raising that rate is worth. U.S. EPA: National overview, facts and figures on materials, wastes and recycling

I completed this lab (wet or virtual)

Lab F: Power iteration and SVD image compression

Week 14

Goal: Find a dominant eigenvalue by repeated multiplication, then compress a small grayscale image with a truncated SVD and see how few numbers keep the picture.

\mathbf{x}_{k+1}=\frac{A\mathbf{x}_k}{\|A\mathbf{x}_k\|},\quad A_k=\sum_{i=1}^k\sigma_i\mathbf{u}_i\mathbf{v}_i^T

Power iteration and truncated SVD

Procedure

In a spreadsheet, start with x₀ = (1, 0) and multiply by A = [[2, 1], [1, 2]] ten times, rescaling each time; watch the ratio of lengths approach 3.
Make an 8 × 8 grid of gray levels (0–255) for a simple picture, or load a photo in free Python with NumPy.
Compute the SVD (numpy.linalg.svd) and rebuild the image with k = 1, 2 and 4 terms.
Record the storage k(8 + 8 + 1) and the share of Σσ² kept, and judge when the picture looks right.

Data table

k	Numbers stored (8 × 8 image)	Share of 64
1	17	26.6%
2	34	53.1%
4	68	106.3%

Exact storage counts. For an 8 × 8 image, k = 4 already stores more than the raw 64 numbers; the savings appear for large images (Unit 10: rank 50 of a 1000 × 1000 image keeps 10.0%).

Sample calculation

Power iteration: (1, 0) → (2, 1) → (5, 4) → (14, 13); ‖(14, 13)‖/‖(5, 4)‖ = 19.10/6.403 = 2.98, near λ = 3.
Storage for k = 1: 8 + 8 + 1 = 17 numbers, 17/64 = 26.6%.

Safety

Take a short screen break every half hour.

Home or virtual alternative

The spreadsheet and free Python versions are both home labs.

MIT OpenCourseWare 18.06 Linear Algebra (Strang, Spring 2010) · Interactive Linear Algebra, 5.1 Eigenvectors

Planet link: NASA JPL’s EMIT returns more than 100,000 spectra every second; compressing and separating that data uses exactly these tools. NASA JPL: EMIT, Earth Surface Mineral Dust Source Investigation

I completed this lab (wet or virtual)

Exam prep practice sets

Time yourself: about 10 minutes per problem. Then compare with the full solutions.

Midterm practice set (Units 1–5)

Find the angle between (1, 0, 1) and (0, 1, 1).
Show full solution
1. Dot = 1; lengths √2 each; cos θ = 1/2.
Answer: 60°.
Solve x + 2y = 4, 3x − y = 5.
Show full solution
1. R₂ − 3R₁: −7y = −7.
Answer: x = 2, y = 1.
Invert [[3, 2], [1, 1]].
Show full solution
1. det = 1.
Answer: [[1, −2], [−1, 3]].
A one-sector economy uses 0.4 of its own output. What output meets demand 120?
Show full solution
1. x = 120/0.6.
Answer: 200.
A 6 × 9 matrix has rank 4. Find dim N(A) and dim N(Aᵀ).
Show full solution
1. n − r and m − r.
Answer: 5 and 2.

Final practice set (all units)

Fit y = C + Dt to (0, 0), (1, 2), (2, 2) by least squares.
Show full solution
1. AᵀA = [[3, 3], [3, 5]], Aᵀb = (4, 6); det 6.
Answer: C = 0.333, D = 1.
Find det [[2, 1, 0], [1, 2, 1], [0, 1, 2]].
Show full solution
1. 2(4 − 1) − 1(2 − 0) + 0.
Answer: 4.
Find the eigenvalues of [[6, 2], [2, 3]].
Show full solution
1. Trace 9, det 14: λ² − 9λ + 14 = 0.
Answer: 7 and 2.
Find the steady state of the Markov matrix [[0.9, 0.2], [0.1, 0.8]].
Show full solution
1. 0.1q₁ = 0.2q₂.
Answer: (0.667, 0.333).
Find the singular values of [[2, 0], [0, −3]].
Show full solution
1. AᵀA = diag(4, 9).
Answer: 3 and 2.
How many numbers does a rank-20 SVD of a 500 × 800 image store, and what share of the full image is that?
Show full solution
1. 20(500 + 800 + 1).
Answer: 26,020, about 6.5%.

Formula sheet

\|v\|=\sqrt{v\cdot v},\ \cos\theta=\frac{v\cdot w}{\|v\|\|w\|}

A\mathbf{x}=\mathbf{b},\ \ A=LU

\begin{bmatrix}a&b\\c&d\end{bmatrix}^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}

(AB)^{-1}=B^{-1}A^{-1},\ (AB)^T=B^TA^T

\mathbf{x}=(I-A)^{-1}\mathbf{d}

r+\dim N(A)=n

\mathbf{p}=\frac{a\cdot b}{a\cdot a}a,\ \ A^TA\hat{x}=A^Tb

A=QR,\ Q^TQ=I

\det(AB)=\det A\det B,\ x_i=\det B_i/\det A

\det(A-\lambda I)=0,\ \sum\lambda=\operatorname{tr}A,\ \prod\lambda=\det A

A=PDP^{-1},\ A^k=PD^kP^{-1},\ A=Q\Lambda Q^T

A=U\Sigma V^T,\ \sigma_i=\sqrt{\lambda_i(A^TA)},\ \kappa=\sigma_1/\sigma_n

Constants

Constant	Value
Speed of light (used for GPS ranges)	299,792,458 m/s
1 mile	1.609344 km
1 foot	0.3048 m
1 inch	25.4 mm
Flops to solve an n × n system by elimination	about n³/3 multiply–adds (Strang)
Identity matrix I	Ones on the diagonal, zeros elsewhere: AI = IA = A

Source: NIST Guide to the SI, Appendix B.9: conversion factors

Free texts and lectures

Use these free textbooks, lectures and tools alongside the course.

MIT OpenCourseWare 18.06 Linear Algebra (Strang, Spring 2010): Strang’s full video lectures, problem sets and exams.
MIT OpenCourseWare 18.06SC Linear Algebra (self-paced, Fall 2011): Self-paced version with recitation videos and worked problems.
Interactive Linear Algebra (Margalit and Rabinoff, Georgia Tech): Free interactive textbook used at Georgia Tech.
Jim Hefferon, Linear Algebra (free text): Free full textbook with answers.
Gilbert Strang, Introduction to Linear Algebra (book site): Strang’s book site with notes and figures.
PhET: Vector Addition: Interactive vectors and components.

Glossary

Basis: A linearly independent set that spans a space.
Column space: All combinations Ax of the columns of A.
Condition number: σ₁/σₙ, how much errors can grow when solving Ax = b.
Determinant: Signed volume scale factor of A; zero exactly when A is singular.
Diagonalization: A = PDP⁻¹ with eigenvectors in P.
Dimension: Number of vectors in any basis.
Eigenvalue: λ with Av = λv for some nonzero v.
LU factorization: A = LU from elimination.
Least squares: The x̂ minimizing ‖b − Ax‖.
Leontief inverse: (I − A)⁻¹, total output needed per unit of final demand.
Linear independence: Only the zero combination gives the zero vector.
Markov matrix: Nonnegative entries, columns summing to 1.
Null space: All x with Ax = 0.
Orthonormal: Mutually orthogonal unit vectors.
Pivot: First nonzero entry of a row in echelon form.
Positive definite: Symmetric with xᵀAx > 0 for x ≠ 0.
Projection: Closest point to b in a subspace.
Rank: Number of pivots; dimension of the column space.
Singular value: σᵢ = √λᵢ(AᵀA).
Span: All linear combinations of a set of vectors.
Subspace: A set containing 0 that is closed under addition and scaling.

Self-assessment rubric

No grade or credit is given. Use this to judge honestly where you are in each unit.

Level	What it looks like
4 · Mastery	Chooses the right tool (elimination, projection, eigenvalues or SVD), computes small cases by hand, checks answers (Ax = b, trace and determinant), and explains the meaning in pictures and in context.
3 · Proficient	Solves the practice problems with occasional arithmetic slips.
2 · Developing	Follows worked examples but mixes up rows and columns or the four subspaces.
1 · Beginning	Manipulates symbols without checking sizes or answers; needs to revisit the unit.

Suggested weighting if you want a score

Part	Weight
Practice problems	25%
Labs	20%
Midterm practice set	25%
Final practice set	30%

Aim for level 3 or better in every unit before moving to the next course.

Linear Algebra

About this course

How this course helps save the planet

Before you start

Prerequisites

How to study

Lab equipment (for the wet labs)

Suggested 15-week schedule

Your progress

Vectors: Linear Combinations, Lengths and Dot Products

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Systems of Linear Equations and Gaussian Elimination

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Matrix Algebra: Multiplication, Inverses and LU Factorization

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Linear Models: Input–Output (Leontief) Economics, Networks and Balancing

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Vector Spaces, Subspaces, Independence, Basis, Rank and the Four Fundamental Subspaces

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Orthogonality, Projections, Least Squares and Gram–Schmidt

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Determinants: Properties, Cofactors, Cramer’s Rule and Volume

Learning objectives

Key concepts

Key equations

Worked examples, step by step