Linear Algebra: Linear Transformations & Vector Spaces

Define linear independence and dependence for a vector set. A dependent set has a vector as a linear combination of the others; independent means only the zero combination exists.

Demonstrates testing linear independence by forming a matrix from vectors, reducing to reduced row echelon form, and showing nontrivial solutions reveal dependence.

Construct a matrix A from the vectors as its columns and compute det(A). Determine linear dependence if the determinant is zero, or linear independence if it is nonzero.

Compare vector count m to the space dimension n; if m > n, the set is linearly dependent. Visualize with reduced row echelon form, where a free column indicates dependence.

Define the span as all linear combinations of a vector set. Show that any z in the span equals c1 v + c2 w, forming a plane in R3.

Define rank as the dimension of the space spanned by vectors, equal to the maximum number of linearly independent columns or rows, computed through reduced row echelon form.

Compute the rank of a 3x3 matrix via row operations to reduced row echelon form, identify two pivot columns, and conclude the rank is two, spanning a plane in R3.

Prove that column rank equals row rank by expressing A as B C, with B containing basis columns, then show row rank of A is at most the column rank.

Complete proof that row rank equals column rank, showing row rank ≤ column rank and column rank ≤ row rank, using a row space basis and A = E D.

Define the column space as the span of a matrix's column vectors. Show that any vector in the column space is a linear combination of the columns A1 through An.

Define the row space as the span of the rows of A and show that the column space of A equals the row space of A transpose.

The null space of a matrix A is x with Ax = 0; if it contains only the zero vector, A is linearly independent, otherwise A is linearly dependent.

Identify a linear subspace as a subset of a vector space that contains the zero vector, is closed under scalar multiplication and addition, and equals the span of its members.

Identify a basis for a subspace by selecting a linearly independent set that spans V within R^n. In R^3, two independent vectors can span a two-dimensional subspace (a plane), illustrating that the subspace's dimension is at most n.

Explore how a subspace can have multiple bases, using a standard basis in R3 and an alternative scaled z basis, and introduce changing coordinates between bases.

Learn how to change basis in R2 using a change of basis matrix, converting a vector from the other basis to standard coordinates by solving for c1 and c2.

Generalize change of basis by constructing the change of basis matrix from basis vectors and solving for a vector's coordinates in a new basis, in any dimension.

Invert the change of basis matrix to move from the new basis to the standard basis and back, using M and M inverse to represent vectors across bases.

Link change-of-basis matrices to linear transformations by showing how to express a transformation in a non standard basis as a' = M^{-1} A M, connecting standard and non standard representations.

Apply the change of basis matrix and its inverse to transform vectors from a non-standard basis to standard coordinates in a 2x2 clockwise rotation example.

Show how to express a rotated vector in a non standard basis by applying m inverse t m, then verify with a linear combination of the new basis vectors.

Learn to recover the standard transformation matrix from a non-standard basis by applying the change-of-basis matrix and its inverse, with a worked example.

Explore the change of basis matrix M, its inverse, and how BTS connects standard and nonstandard basis transformations.

Reconstruct the original transformation matrix with respect to the standard basis using a change-of-basis matrix, its inverse, and a D matrix built from transformed vectors, via a 90-degree rotation example.

Explore how Gram-Schmidt converts a set of linearly independent vectors into an orthonormal basis, using projections and normalization to form orthogonal, unit-length vectors.

Explore eigenvalues and eigenvectors of a matrix, showing how A v = lambda v scales vectors by lambda and how det(lambda I - A) = 0 characterizes these pairs.

Compute eigenvalues for a 2x2 matrix by forming the characteristic equation det(lambda I minus A) = 0; with A = [[2,1],[3,4]], eigenvalues are 1 and 5.

Explain why solving lambda I minus A yields eigenvalues by linking nontrivial eigenvectors to a singular matrix, indicating linear dependence of columns and a zero determinant.

Compute eigenvectors for a 2x2 matrix by solving (λI−A)v=0 for eigenvalues 5 and 1; obtain eigenvectors proportional to (1,1) and (3,1) via row reduction.

Visualize the eigenvalues and eigenvectors of a 2x2 matrix on a coordinate plane, with eigenspaces spanned by (3,1) for lambda=1 and (1,1) for lambda=5.

Compute eigenvalues of a 3x3 matrix by solving det(lambda I minus A) = 0, using the Sarrus rule, factoring the cubic, and identifying eigenvalues 1 and 3.

Solve (lambda I - A) v = 0 to find eigenvectors of the 3×3 matrix. For lambda=1, v ∝ [1, -2, 1]; for lambda=3, v ∝ [1, 0, 1].

Verify eigenpairs by showing that A v = λ v holds for general eigenvectors, using symbolic c, for λ = 1 and λ = 3.

Diagonalization expresses A as P D P inverse, with P containing eigenvectors and D the diagonal of eigenvalues; A is diagonalizable if it has n linearly independent eigenvectors.

An eigenbasis is a set of eigenvectors that forms a basis for subspace; distinct eigenvalues ensure independence, and using this basis yields a diagonal matrix with eigenvalues on its diagonal.

We define abstract vector spaces as sets with a zero element closed under addition and multiplication by real numbers, obeying additive, distributive, and multiplicative axioms, moving beyond vectors as arrows.

View functions as vectors by treating all functions as a vector space closed under addition, inverses, and scalar multiplication, and see polynomials expressed via a basis and coordinates.

Explore abstract vector spaces by treating gremlins as vectors, showing how functions can form a vector space and how linear operations generalize problems.