Linear Algebra and Geometry 2

Explore moving from abstract to concrete by defining a function as a rule between sets x and y with exactly one value per input, using graphs and one-variable examples.

Transition from concrete to abstract by starting from R^n and defining vector spaces, subspaces, linear combinations, span, linear independence, basis, dimension, and coordinates.

Recalls vectors in R2, R3, and RN as lists of coordinates like delta x, delta y, delta z, and demonstrates coordinate-wise addition and scaling, preparing prototype for vector space concept.

Formulate the formal definition of vector spaces using axioms and examine R^n as a primary example, focusing on addition and real-number scaling.

Explore how the set of m by n real matrices forms a vector space under element-wise addition and scalar scaling, with the zero matrix as the neutral element.

Explore how the functions from an interval to real numbers form a vector space under pointwise addition and scaling, with the zero function and additive inverses.

Complex numbers form a vector space under standard addition and real-scalar multiplication, with closure, commutativity, and associativity, plus zero and additive inverse as axiom checks.

Derive the cancellation property in vector spaces using only axioms: apply additive inverses, associativity, and the zero vector to show u equals w when v+u = v+w.

Prove uniqueness of the additive identity and inverses, and define the difference u − v as u plus the inverse of v, unique solution to u = x + v.

Derives core vector space properties from axioms: zero times v is zero; scalar times zero is zero; minus one times v is the additive inverse; and the zero product property.

Identify subspaces as vector-space subsets closed under addition and scaling, illustrated by lines through the origin in R2, and note that their intersection is also a subspace.

Explore subspaces of R2, including lines through the origin, the trivial subspace, and the whole space, noting closure under addition and scalar multiplication.

Explore the subspaces of R3, including lines through the origin, planes through the origin, the trivial subspace, and the whole space, and distinguish affine subspaces not through the origin.

Assess subspaces in R3 by the zero vector, closure under addition, and scaling; the x-axis and the plane x-y+z=0 are subspaces, while the line y=1 is not.

Investigate which sets of functions form subspaces, including continuous and discontinuous functions, polynomials, and even and odd function spaces, with proofs of closure under addition and scaling.

The lecture examines subspaces of the real n by n matrices. It shows diagonal and symmetric matrices are subspaces, while determinant-one and trace-one sets are not; trace-zero forms a subspace.

The set of vectors in R^n orthogonal to u0, by u0 · v = 0, forms a subspace with zero vector and closure under addition and scalar multiplication.

Explore linear combinations and linear independence through a favorite non-singular 2x2 matrix, illustrating determinants, inverses, Gaussian elimination, Gauss-Jordan elimination, linear transformations, and Cramer's rule.

Define linear combinations in any vector space, and illustrate with standard basis e1, e2, e3 in R^n, showing how vectors arise as coordinate sums and parallelogram rule.

Explore linear combinations in vector spaces by forming new matrices as sums of scaled basis matrices, illustrated with 2x2 real matrices and the symmetric case.

Solve whether given vectors are linear combinations of two base vectors via an overdetermined system for c1 and c2, and interpret geometrically whether they lie in plane through origin.

Express A, B, C, and D as linear combinations of u, v, and w in R3 by solving four 3×3 systems via Gaussian elimination, guaranteed by a nonzero determinant.

Understand the span of vectors as all linear combinations, the smallest subspace containing the given vectors; a line, a plane, or R3 arise from one, two, or three vectors.

Determine if v lies in the span of five vectors in R4 by solving an augmented system; Gaussian elimination shows solvability only when a equals minus two.

Gaussian elimination shows the system is consistent for all a. Therefore, v lies in the span for every value of a.

Determine for which a the vector v lies in M, the span of five vectors in R4, since all spanning vectors have last coordinate zero while v’s is one.

Explore the meaning of trivial in linear algebra through homogeneous systems, zero solutions, and determinant-based uniqueness. Examine linear independence and dependence via zero and nonzero linear combinations.

Define linear independence and dependence via vector equations and linear combinations. Relate solvability of homogeneous systems, determinants, and matrix representations to when vectors are independent or dependent.

Explore the geometry of linear independence and dependence in R2 and R3, linking solvability and uniqueness of homogeneous systems with two non-parallel vectors independent and three coplanar vectors dependent.

Explains linear independence in r^n and how reduced row echelon form reveals independent rows and columns, with pivot positions marking independence and elementary row operations preserving the span.

Transform the matrix to row reduced echelon form to identify a linearly independent generating set for the row vectors, and justify independence and spanning via linear combinations.

Analyze linear independence in the space of symmetric 3×3 matrices (problem 7), deriving a linearly independent generator set and a decomposition using x, y, z coefficients.

Demonstrate linear independence of the polynomials 1, t, t^2 in the space of polynomials of degree ≤ 2, and show that 1, t, and 4-t are linearly dependent.

Generalize the Vandermonde determinant for n distinct numbers, show its nonzero value, and prove that {1, t, t^2, ..., t^(n-1)} is linearly independent and generates polynomials of degree ≤ n-1.

Proves that e^t, e^{2t}, and e^{3t} are linearly independent in the space of smooth functions on R by differentiating a linear combination and using a nonzero Vandermonde determinant.

Learn how the wronskian determines linear independence of smooth functions in C∞(R) by differentiating and building a determinant, illustrated with the polynomials 1, t, t^2, t^3 whose wronskian equals 12.

Show that the set {1, cos t, sin t} is linearly independent in the space of smooth functions by computing its Wronskian and using the Pythagorean identity.

Show that the n exponential functions e^{λ_i t} with distinct λ_i are linearly independent in the space of smooth functions on R, by a nonzero wronskian and Vandermonde determinant.

A basis is a linearly independent set that spans a vector space, so V equals the span of its basis vectors, and the dimension equals the number of basis vectors.

This lecture demonstrates when three vectors in R3 form a basis, using the determinant test and the Sarrus rule, and linear independence, with lambda-dependent cases solved via polynomial roots.

Discover how bases in the plane and in three space relate to linear independence, with determinant tests, parallelepiped volume, and the equivalence of many linear-algebra statements.

Two given vectors in R3 extend to a basis with w by the determinant test, which yields -19 w1 - w2 + 7 w3 ≠ 0.

Solve problem three in four-space by finding the intersection of subspaces m and n, yielding a two-parameter form and a two-vector basis for m ∩ n.

Find the intersection of two four-dimensional subspaces by equating linear combinations of the span vectors, solving a homogeneous system with Gaussian elimination, and deriving a basis of two independent vectors.

Determine a basis for the subspace spanned by four polynomials in P3; f1, f2, f4 form the basis, and adding f5(x)=1 completes the basis for P3.

Explore coordinates with respect to a basis, expressing vectors as coefficients in that basis rather than standard axes. A basis spans the space and is linearly independent, yielding unique representations.

coordinates with respect to a basis are unique because a basis is linearly independent and spans the space; every vector has a unique linear combination of basis vectors.

Explore coordinates with respect to a basis via a unifying example, where coordinates are coefficients of column vectors and depend on the chosen basis, determinant nonzero confirming a valid basis.

The dimension of this subspace in R4 is 3, since v3 equals -2 v2, leaving v1, v2, v4 as a basis.

Construct a basis for the space of functions from a four-element set to R and determine coordinates of a function in this basis using basis functions b1 to b4.

Examine change of basis and coordinate calculation in two dimensions, using visual illustrations and multiple methods to convert vector coordinates from the standard basis to a new basis.

Compute coordinates of a vector in a nonstandard basis of R3 by solving the linear system, verify the basis via determinant, and introduce the transition matrix for changing coordinates.

Derive and apply the transition matrix to convert coordinates between new and old bases, using columns as new basis coordinates in the old basis and its inverse to return.

Use a transition matrix to convert coordinates from the new basis to the old basis and back, and apply its inverse to recover the new basis coordinates.

Illustrate how to use a transition matrix to convert coordinates from the basis f1, f2 to the standard basis in R2, using vectors v1–v4 and the parallelogram reading rule.

Compute the coordinates of a vector in a new basis using the transition matrix and its inverse, connecting standard basis coordinates to the new basis and visualizing via a parallelogram.

Learn to compute the transition matrix between two non-standard bases using method one, by expressing the new basis vectors in terms of the old basis with coordinates and geometric intuition.

Explore a general method to find the transition matrix between two non-standard bases by passing through the standard basis, using easy-to-build matrices and their inverse.

Compute the transition matrix between two non-standard bases using Gaussian elimination and elementary matrices, by forming a double matrix in the standard basis.

Explore change of bases in linear algebra by constructing transition matrices between non-standard bases, applying Gaussian elimination, and computing coordinates of vectors across B, B' and the standard basis.

Apply two methods to find coordinates of a vector in a basis: solving a linear system and using a transition matrix, demonstrated in r2 with standard and orthogonal bases.

Verify that B and C are bases in R3 by determinant test; compute R equals u minus 2v plus w and obtain its coordinates in B and C via elimination.

Solve a change of basis problem in r2 by using the known transitions b1→b2 and b2→b3, then compute b3→b1 as the product of their inverses.

Compute the coordinates of a degree-two polynomial relative to the given basis in a change-of-basis problem, solving a linear system via Gaussian elimination to get (5, 1, -2).

Discover how to change to an orthonormal basis in the plane, use rotation matrices, and relate coordinates across bases through the transition matrix, with determinant one and inverse equals transpose.

This section introduces matrices and develops bases for row space, column space, and null space, along with kernels and images of linear transformations.

Learn to identify the row space and column space as spans of rows and columns, with rows as vectors in R^n and columns in R^m, and determine their bases.

Row reduce a matrix to a row-reduced echelon form to extract a basis for the row space from pivot rows, using Gaussian elimination.

Learn how elementary row operations affect the column spaces and determine a basis for the column space by row reducing to reduced row echelon form, using leading-one columns.

Row reduce to row echelon form and pick the pivot columns; their corresponding original columns form a basis for the column space, while dependence relations stay consistent under row operations.

Determine a basis for the span of four vectors in R5 by forming a matrix and applying row reduction to reveal the row space basis and the dimension is three.

Form a matrix with the given vectors as columns, then compute a column-space basis to select a subset that spans subspace; express v as a combination of u1, u2, u4.

Identify a basis for the span of five vectors in R4 using Gaussian elimination, selecting u1, u2, and u5 as a basis, and compute each vector's coordinates in that basis.

Transpose the matrix to turn rows into columns. Then row-reduce the transposed matrix to identify leading columns, whose corresponding original rows (R1, R2, R4) form the row space basis.

Treat degree-three polynomials as vectors in the standard basis, form a 4×5 matrix of coordinates, and use Gaussian elimination to obtain a basis from p1, p2, p4 and their coordinates.

learn how the null space of a matrix is the solution set to A x = 0, forming a subspace of RN, in contrast to column and row spaces.

Learn to find the null space of a matrix by row reducing to reduced row echelon form, identifying free variables, and constructing a basis from parametric solutions.

Derive the nullspace basis and its dimension by reducing the augmented matrix of the homogeneous system to row-reduced form and identifying free variables as parameters.

Demonstrates finding a basis for the null space of A via row reduction and solving a non-homogeneous system, then explains the affine line of solutions parallel to the homogeneous one.

Define the rank of a matrix as the dimension of its column space and the number of pivot positions in row reduced form, linking it to row space and transpose.

Define nullity as the dimension of the null space of a matrix. Relate nullity to rank by showing a matrix with two pivot columns and a nullity of four.

Explore the rank–nullity relationship for an m by n matrix, where rank equals the dimension of the column space and nullity equals the dimension of the null space.

explain the rank–nullity relation: rank plus nullity equals the number of columns, and rank does not exceed min(rows, columns); for 4×6, 5×5, 6×4, max ranks are 4, 5, 4.

Explore the relationship between rank and nullity in solving ax=b, using Gauss-Jordan to show how coefficient and augmented ranks determine consistency and parameter count.

Explore the rank–nullity relationship for the given matrix, showing rank one is impossible and rank two only occurs when r=2 and s=1; otherwise the matrix has rank three.

Explore orthogonal complements of subspaces, define the orthogonal complement, and relate it to the null space and row space via homogeneous systems and the fundamental theorem of linear algebra.

Explore the four fundamental matrix spaces of an m by n matrix: column space, row space, null space, and the null space of A transpose, and their roles in solvability.

Explore the fundamental theorem of linear algebra, including the column space, row space, null space, and null space of A^T, rank-nullity relations, and orthogonal complements with direct sum decompositions.

Learn linear transformations in linear spaces and how linearity preserves linear combinations. See why sine, logarithm, and exponential are non-linear, and anticipate the arithmetical section before illustrations and animations.

Explore the basic terminology of functions and linear transformations, including domain, codomain, and range (image). Identify surjections, injections, and bijections, and relate linear maps to operators, isomorphisms, and endomorphisms.

Learn to think about functions from R^n to R^m by viewing inputs as n coordinates and outputs as m coordinates, with component functions shaping each coordinate.

Use approach one to decide when a function from R^n to R^m is linear: each component is a linear combination of inputs with coefficients; zeros allowed.

Discover approach two: linear transformations from R^n to R^m are matrix transformations, realized by A x, with standard matrices and examples.

Demonstrate approach three: a linear transformation from R^n to R^m preserves sums and scalar multiples, mapping linear combinations to linear combinations, with examples and a note on constants breaking linearity.

Show that approaches two and three are equivalent for describing linear transformations from R^n to R^m, using matrix multiplication and the standard basis to form the matrix.

Compute t(102) for a linear transformation from R3 to R3 using direct substitution and the standard matrix. Confirm both methods yield the same result, t(102) = (6, 12, -3).

Explore image and kernel of a linear transformation and their role in invertibility; relate image to column space and kernel to null space, then solve a 3 by 3 system.

Construct the standard matrix A for a linear transformation from R4 to R3, row reduce to reveal the image (column space), and select the leading columns as a basis.

Compute the kernel of a linear transformation from R4 to R3 by solving the homogeneous system via Gauss elimination, obtaining a parametric solution with two basis vectors and dimension two.

Determine the image and kernel of a linear transformation from R4 to R2 by building the 2x4 matrix, finding the null space and column space.

Explore inverse operators in linear algebra, including determining invertibility of 2x2 operators, computing inverse matrices, and relating kernels and images.

Determine the standard matrix of a linear transformation from R2 to R2 using the images v1→w1 and v2→w2, by solving A [v1 v2] = [w1 w2].

Explore kernel and geometry of a linear transformation from R3 to R2, determine the kernel, null space, and its one-dimensional line spanned by [0, -1, 1].

Study linear transformations from R4 to R3 using linearity and two known images T(u1) and T(u2). Determine whether v is a linear combination of u1 and u2 to find T(v).

Explore linear transformations from R^n to R^m using matrix representations and change of basis. See how standard grid lines transform into a parallelogram via the matrix columns t(e1) and t(e2).

This lecture analyzes a linear operator from R2 to R2 with a nontrivial kernel, showing image is a line spanned by (3, -5) and the kernel is the y-axis.

Explore line symmetries in the plane by defining symmetry axes, fixed points on the axis, and reflecting points across perpendicular lines; learn matrix representations, identity transformation, and isometries in r2.

Compute the orthogonal projection of vectors in three-dimensional space onto a given vector u and the corresponding projection matrix 1/14[[1,2,3],[2,4,6],[3,6,9]], whose image is span{(1,2,3)} and kernel is the plane x+2y+3z=0.

Learn how to compute the standard matrix for reflection about the line y = 2x through the origin using projection onto the line’s direction, generalizable to lines y = kx.

Rotate the plane about the origin by 90 degrees using a matrix that sends e1 to e2 and e2 to -e1, preserving distances; its inverse is rotation by -90 degrees.

Construct the standard matrix for rotating the plane about the origin by angle alpha using e1 and e2, showing the cosine–sine form, determinant one, and inverse rotation by minus alpha.

Explore linear transformations in R2 through expansions, compressions, scaling, and shear, with animations showing matrices action on vectors, grid lines, and resulting parallelograms.

Learn plane symmetry in three-space by reflecting across the coordinate planes, derive corresponding 3×3 transformation matrices, and apply them to vectors like (3,4,5).

Explore orthogonal projections onto the coordinate planes in three-space, derive projection matrices for xy, yz, and xz planes, and apply them to a vector like (3,4,5).

Compute the standard matrix for symmetry about a plane through the origin in three-dimensional space by subtracting twice the projection on the plane's normal, proving a linear transformation.

This lecture derives a standard matrix for orthogonal projection onto a plane through the origin with normal (3,2,1), identifying the image as the plane 3x+2y+z=0 and kernel as span(3,2,1).

Apply matrix multiplication to rotate vectors about the x, y, and z axes, using the right-hand rule and cosine and sine values for unit vectors E1, E2, E3.

Explore how vector and affine subspaces transform under linear transformations, while examining area and volume changes in R2 and R3 and composing transformations.

Explore how linear transformations map vector subspaces and affine subspaces, preserving linear combinations and mapping the zero vector to zero, so lines and planes become lines, planes, or points.

Explore how a non-singular matrix maps parallel lines to parallel lines in R2, using parametric descriptions, directional vectors, and basis changes to reveal preserved linearity.

map a line under an invertible matrix using intercept-slope form to obtain y' = 4/5 x' + 1/5, and compare this with the parametric approach.

Examine how linear transformations from R2 to R2 and R3 to R3 alter area and volume of parallelograms and parallelepipeds, with determinants scaling and sometimes reversing orientation.

Discover how a linear transformation maps the unit disk to an ellipse with semi-axes a and b, giving area pi a b and the equation x^2/a^2 + y^2/b^2 = 1.

Explore compositions of linear transformations, where the order matters and the first performed is last. Understand how domain and departure and target sets govern possible compositions.

Learn how to obtain the standard matrix of a composition of linear transformations by multiplying their standard matrices, and see that the order matters.

Explain why the composition of linear transformations is linear and how its standard matrix equals the product of the two standard matrices. Learn how matrix multiplication mirrors transformation composition.

Explore compositions of linear transformations from R2 to R4 by computing the standard matrices for T and the pi/6 rotation, and explain why T∘R is possible while R∘T is not.

Demonstrate composing linear transformations in R^2 to derive standard matrices for three problems, using reflections, projection onto the y-axis, rotations, and a dilation, with animation.