Become a Linear Algebra Master

Explore how matrices organize data and simplify solving linear systems, from two unknowns to large equation sets, and learn the basics of using matrices in linear algebra.

Explore how matrices solve two-variable linear systems, linking substitution, elimination, and graphing to the same solution, and see why matrices simplify larger problems.

Learn to solve a linear system of three equations in three unknowns using elimination and substitution, without matrices, and explore unique, no solutions, and infinite solutions.

Explore how matrices organize data to express systems of linear equations, define dimensions in rows by columns, and identify entries with lowercase indices using examples A, B, C, and D.

Represent linear systems with matrices by organizing coefficients into a coefficient matrix and augmenting with constants to form an augmented matrix, aligning x, y, z by columns.

Master simple row operations—switching rows, scaling rows, and adding rows—to transform augmented matrices and solve linear systems through elimination.

Learn to convert matrices to row echelon form and reduced row echelon form, using pivot entries, a staircase pattern, and bottom-placed zero rows to read solutions directly.

Gauss-Jordan elimination becomes a repeatable algorithm that transforms a matrix into reduced row echelon form by pivoting and zeroing, solving linear systems such as x=1, y=2, z=3.

Learn how to determine whether a linear system has one, none, or infinitely many solutions using Gauss Jordan elimination, identifying pivot and free variables in the reduced row echelon form.

Explore matrix addition, subtraction, and multiplication, and learn when these operations are defined, using zero and identity matrices for real-world problems like computer graphics.

Learn to add and subtract matrices with matching dimensions, using corresponding entries; note that addition is commutative and associative, while subtraction is not, then solve matrix equations.

Explore scalar multiplication of matrices and rows, including distributing scalars like 2, 1/2, 1, and 0 across entries, and solving for unknown matrix Y.

Identify the zero matrix as a matrix of all zeros, distinguished by its dimensions. Show that opposite matrices sum to the zero matrix when adding a matrix to its negative.

Multiply matrices using dot products, not entrywise, by ensuring the inner dimensions match; the result has outer dimensions, and matrix multiplication is not commutative but is associative and distributive.

Explore identity matrices, which have a diagonal of ones and zeros elsewhere, and are always square. Understand that multiplying by I_n leaves a matrix unchanged, preserving dimensions.

Learn how a square matrix A can become the identity via a single elimination matrix E, equivalent to the inverse, by bundling Gauss Jordan row operations into E.

Explore vectors as directional quantities defined by length and direction, distinct from matrices. Learn vector arithmetic, linear combinations, linear independence, and the ideas of spaces and basis vectors.

Explore vectors as direction and magnitude, how they map to matrices as column or row vectors, and how standard position, initial and terminal points illustrate their geometry in n-dimensional spaces.

Learn basic vector operations: addition, subtraction, and the dot product, and how vectors relate to matrices, including graphical intuition and commutativity of addition.

Find unit vectors in same direction as a given vector and normalize its length to one. Express vectors as a linear combination of I, J, K in R2 and R3.

Demonstrate expressing vectors as linear combinations of standard basis, define span as all such combinations, and explain how two independent vectors span R2 while three independent vectors span R3.

Demonstrate that two linearly independent vectors in R2 span the plane. Test independence by solving c1*v1 + c2*v2 = 0 to see if only zero solution exists.

Learn how to test linear independence in three dimensional space by forming a matrix from three vectors and using Gauss-Jordan elimination to check for only the trivial solution to c1*v1+c2*v2+c3*v3=0.

Learn how linear subspaces, such as all vectors in R^2 with x+y=0, are defined by containing the zero vector and being closed under addition and scalar multiplication.

the span of a vector set is always a subspace, closed under addition and scalar multiplication, and contains the zero vector.

Define basis as a spanning, linearly independent vector set that defines a subspace, with two vectors in R2 forming a basis for that space.

Explore how dot and cross products reveal the relationships between vectors in two- and three-dimensional space; the dot product measures alignment, while the cross product length captures directional difference.

Master dot products for vectors and matrices, compute A · B in any dimension, relate vector length to A · A, and recall commutative, distributive, and associative properties.

Explore the Cauchy–Schwarz inequality, linking the absolute dot product to the product of vector lengths. Learn when equality occurs for scalar multiples, and test linear independence versus dependence with examples.

Derive the vector triangle inequality from the Cauchy–Schwarz inequality to relate vector lengths and test linear independence, illustrated by a four-dimensional example.

Explore how the dot product and vector lengths determine the angle between vectors, identify orthogonality via zero dot product, and compute angles using cosine theta with real examples.

Define planes in three-dimensional space from a normal vector and a vector in the plane. Derive the plane's equation Ax+By+Cz=D using a point and the normal, via the dot product.

Compute the cross product of two 3D vectors to obtain a vector orthogonal to both, via a determinant with I J K; use the right-hand rule for direction and magnitude.

Compare how the dot product measures alignment of vectors, and how the cross product captures their directional difference. Highlight orthogonal, parallel, and opposite directions.

Explore how matrix-vector products reveal the four fundamental subspaces—null space, column space, row space, and left null space—and solve for the null and column spaces.

Multiply a matrix by a vector using the same rules as matrix multiplication, treating the vector as a column and obtaining an m by one result, with dimensions matching.

The null space of a matrix, defined by A x = 0, is a subspace that always includes the zero vector and reveals whether the columns are independent.

Compute the null space of a 3 by 4 matrix by reducing to reduced row echelon form, identifying pivot and free columns, and expressing the solution as the span of [0,1,0,0] and [-1,0,0,1].

Explore the column space as the span of A’s columns and its link to Ax=b; learn to find a basis by testing independence and using the null space.

Solve ax=0 to obtain the complementary (null space) solution and ax=b to obtain the particular solution; add them to form the general solution.

Explore how the dimension of a vector space equals the number of basis vectors, and how nullity and rank determine the null space and column space of a matrix.

Explore how transformations move vectors to new locations and can be represented as matrices, combining reflections, stretches, and rotations into one transformation matrix. Later, study linear transformations, projections, and compositions.

Explore how transformations extend functions to vectors, mapping from a domain to a code domain, and distinguishing range from domain in vector spaces.

Describe how a transformation matrix maps the standard basis i and j, transforms any vector via multiplication, and converts the pre-image to the image of a subset.

Explores how a transformation maps vectors from subset a to subset b, and defines the image, the preimage, and the kernel as all vectors mapping to the zero vector.

The lecture shows that linear transformations map vectors via a matrix-vector product and preserve addition and scalar multiplication, expressible by a matrix whose columns are standard basis images.

Rotate vectors as linear transformations using 2d rotation matrix with theta, cos theta, -sin theta, sin theta, cos theta; in 3d, apply rotation matrices about x, y, or z axes.

Learn to add linear transformations by adding their matrices and applying to vectors, and to scale a transformation by a scalar via its matrix.

Explore projecting onto a line as a linear transformation by constructing a projection matrix A for the line L spanned by (2,1) and using A to project onto L.

Explore compositions of linear transformations between spaces A, B, and C, showing how S and T combine into a single matrix C via t∘s, with matrix-vector products.

Explore inverses in linear algebra, including inverse transformations and matrix inverses that undo actions to yield the identity, and discuss when inverses exist and how to solve systems, noninvertible cases.

Discover how the inverse of a transformation exists and is unique, and how identity transformations, one-to-one, and onto properties govern mapping from A to B and back.

Understand that a transformation written as A x is invertible only when A is square and maps within the same space. Rectangular A cannot be invertible.

Explore why the inverse of a linear transformation is linear, closed under addition and scalar multiplication, and can be expressed as the inverse matrix times the vector.

Explore how matrix inverses act as division, using the determinant to test invertibility and applying the 2x2 inverse formula to detect singular matrices.

Solve a linear system by multiplying the inverse of the coefficient matrix by the right-hand side vector. Compute the inverse via determinant formula or augmented form, then get x and y.

Explore determinants of square matrices, learn methods to calculate them, and understand rules governing determinants. Apply determinants to find the area between two vectors.

Compute determinants by breaking down into two by two determinants with ad minus bc, use row or column choices for larger matrices, apply checkerboard pattern and the rule of Saurus.

Use cramer's rule to solve linear systems via determinants. Replace columns to form Dx, Dy, and Dz, divide by D, and obtain x, y, z in 2×2 and 3×3 examples.

Explore how scaling a matrix row affects its determinant, why duplicate rows force zero, and how row swaps change the determinant’s sign while row operations leave it unchanged.

Explore upper and lower triangular matrices, learn how zeroing below or above the main diagonal simplifies determinant calculation, and use row operations to transform any matrix to triangular form.

Determinants help find the area of transformed figures by treating matrix columns as adjacent sides of a parallelogram, with area scaled by the absolute value of the transformation's determinant.

Define the transpose by swapping rows and columns, then show how to calculate it. Explore how determinants and the four fundamental sub spaces relate between a matrix and its transpose.

Learn how to form a transpose by swapping rows and columns, how dimensions flip, and how the determinant of a transpose equals the original determinant when defined.

Master how transposes interact with products, sums, and inverses, using the key rules (AB)^T = B^T A^T, (A+B)^T = A^T + B^T, and (A^T)^{-1} = (A^{-1})^T.

Compute the transpose and use reduced row echelon form to identify the row space and left null space of the transpose.

Explore the product of a matrix and its transpose, showing that A^T A is always square and invertible when A’s columns are linearly independent.

Decompose a into l u, with l lower triangular with ones on the diagonal and u upper triangular, using elimination matrices and row operations.