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Linear Algebra course

Understand vector spaces, matrices, linear maps, and eigenstructure.

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7Core units
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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 14-week schedule.

Linear Algebra course map: 7 units
#UnitWhat you can do after itWeeks
01Systems and eliminationYou can turn a system into a matrix and solve it with row operations.Weeks 1-2
02Matrix algebraYou can multiply matrices, find an inverse, and use a matrix to move vectors around.Weeks 3-4
03DeterminantsYou can compute a determinant and use it to say whether a matrix can be inverted.Weeks 5-6
04Vector spacesYou can test whether a set of vectors spans a space, is independent, and forms a basis.Weeks 7-8
05Linear transformationsYou can write a transformation as a matrix and find its image and its kernel.Weeks 9-10
06EigenvaluesYou can find the directions a matrix only stretches, and by how much it stretches them.Weeks 11-12
07OrthogonalityYou can project one vector onto another, build a perpendicular basis, and fit a least-squares line.Weeks 13-14

Linear Algebra course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

01

Systems and elimination

Weeks 1-2

You can turn a system into a matrix and solve it with row operations. You can also tell whether it has one answer, no answer, or infinitely many.

  • Row reduce an augmented matrix to echelon form.
  • Read a row of all zeros with a nonzero right side as no solution.
  • Write an infinite solution set using a free variable.

Worked example

Solve x + y = 5 and 2x + 3y = 13

  1. Row 2 minus 2 times row 1: y = 3
  2. Back-substitute: x + 3 = 5

Answerx = 2, y = 3

Most common mistakeSubtracting the rows without scaling first: row 2 minus row 1 gives x + 2y = 8, which still has two unknowns and stalls the elimination.

Explain systems and eliminationPractice systems and elimination
02

Matrix algebra

Weeks 3-4

You can multiply matrices, find an inverse, and use a matrix to move vectors around. You can also say when a product is even allowed.

  • Check that the inner dimensions match before you multiply.
  • Invert a 2 by 2 matrix with the ad minus bc formula.
  • Solve Ax = b by computing the inverse of A times b.

Worked example

Find the inverse of the matrix with rows (4, 7) and (2, 6)

  1. Determinant = 4(6) - 7(2) = 10
  2. Swap 4 and 6, negate 7 and 2, then divide everything by 10

AnswerRows (0.6, -0.7) and (-0.2, 0.4)

Most common mistakeInverting each entry on its own: rows (0.25, 0.143) and (0.5, 0.167) do not multiply back to the identity matrix.

Explain matrix algebraPractice matrix algebra
03

Determinants

Weeks 5-6

You can compute a determinant and use it to say whether a matrix can be inverted. You can also read it as how much the matrix stretches area or volume.

  • Expand a 3 by 3 determinant along a row that contains a zero.
  • Use a determinant of zero to detect a matrix with no inverse.
  • Explain that a determinant of -2 doubles area and flips orientation.

Worked example

Find the determinant of the matrix with rows (3, 1) and (6, 2)

  1. Determinant = 3(2) - 1(6)
  2. 6 - 6

Answer0, so the matrix has no inverse

Most common mistakeAdding the two diagonal products instead of subtracting: reporting 12 and wrongly declaring an invertible matrix, when the rows are actually multiples of each other.

Explain determinantsPractice determinants
04

Vector spaces

Weeks 7-8

You can test whether a set of vectors spans a space, is independent, and forms a basis. You can also find the dimension of a subspace.

  • Test independence by row reducing the vectors placed as columns.
  • Find a basis for the null space of a matrix.
  • Use rank plus nullity equals the number of columns.

Worked example

Are (1, 2) and (3, 6) independent?

  1. Is one a multiple of the other? 3 times (1, 2) is (3, 6)
  2. Yes it is, so they are dependent

AnswerThey span only a line, so the space they span has dimension 1

Most common mistakeCalling two vectors independent because they look different: (1, 2) and (3, 6) sit on exactly the same line through the origin.

Explain vector spacesPractice vector spaces
05

Linear transformations

Weeks 9-10

You can write a transformation as a matrix and find its image and its kernel. You can also change the basis you are describing it in.

  • Build a matrix by tracking where each basis vector lands.
  • Find the kernel by solving Ax = 0.
  • Write a 90 degree rotation as a 2 by 2 matrix.

Worked example

A transformation sends (1, 0) to (0, 1) and (0, 1) to (-1, 0). Write its matrix.

  1. Where the basis vectors land become the columns
  2. First column (0, 1), second column (-1, 0)

AnswerRows (0, -1) and (1, 0), a 90 degree counterclockwise rotation

Most common mistakeWriting the images as rows instead of columns: the resulting matrix sends (1, 0) to (0, -1), rotating the wrong way round.

Explain linear transformationsPractice linear transformations
06

Eigenvalues

Weeks 11-12

You can find the directions a matrix only stretches, and by how much it stretches them. You can also use them to raise a matrix to a high power cheaply.

  • Solve the determinant of (A minus lambda I) equals 0.
  • Find an eigenvector by solving (A minus lambda I)x = 0.
  • Diagonalize a matrix when it has enough independent eigenvectors.

Worked example

Find the eigenvalues of the matrix with rows (2, 1) and (1, 2)

  1. Determinant of (2 - lambda, 1; 1, 2 - lambda) = (2 - lambda) squared - 1
  2. (2 - lambda) squared = 1, so 2 - lambda is 1 or -1

Answerlambda = 1 and lambda = 3

Most common mistakeSubtracting lambda from only the first diagonal entry: solving (2 - lambda)(2) - 1 = 0 gives lambda = 1.5, which is neither eigenvalue.

Explain eigenvaluesPractice eigenvalues
07

Orthogonality

Weeks 13-14

You can project one vector onto another, build a perpendicular basis, and fit a least-squares line. You can also check whether a matrix is orthogonal.

  • Project one vector onto another with the dot-product formula.
  • Run Gram-Schmidt on two independent vectors.
  • Solve the normal equations for a best-fit answer.

Worked example

Project v = (3, 4) onto u = (1, 1)

  1. v dot u = 3 + 4 = 7 and u dot u = 1 + 1 = 2
  2. Scale u by 7/2

Answer(3.5, 3.5)

Most common mistakeDividing by the length of v rather than by u dot u: reporting (1.4, 1.4), which is not the closest point on the line through u.

Explain orthogonalityPractice orthogonality

Prepare for Linear Algebra practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Systems and elimination from a blank page. Then connect it to Matrix algebra. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The Matrix calculator can test calculations and representations used in Linear Algebra. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Linear Algebra practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesLinear Algebra explainedConcepts, worked methods and checks behind this course. PracticeMixed Linear Algebra questionsAnswer review after every attempt. ReferenceFormulas for this courseConditions and a worked example for each one.
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Assess

Take the complete Linear Algebra test

Begin test →Reference

Review essential formulas

Open library →Calculate

Use the Matrix calculator

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Prepare around your exam date

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Questions about the Linear Algebra course

Where should I start?

Start with Systems and elimination if you are following the full sequence. If that unit feels automatic, open the Linear Algebra practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Linear Algebra test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The Matrix calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.