Math 214

Definitions to Know

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Flashcard Homework

To complete flashcard homework, you must make flashcards out of the definitions/theorems below. When collected, these will be examined for completion.

Definitions

Know these word-for-word from the lecture notes/text.

• Chapter 1
  1. Solution (to linear equation)
  2. Inconsistent System
  3. Consistent System
  4. Reduced Row Echelon Form
  5. Homogeneous System
  6. Trivial Solution
  7. Equal (Matrices)
  8. Linear Combination
  9. Matrix Multiplication (formula for c_ij)
  10. Transpose
  11. Trace
  12. Invertible Matrix
  13. Elementary Matrix
  14. Diagonal Matrix
  15. Upper/Lower triangular matrices
  16. Symmetric Matrix
• Chapter 2
  1. eigenvalue/eigenvector
  2. characteristic equation
• Chapter 3
  1. equivalent vectors
  2. norm of a vector
  3. unit vector
  4. Dot product: cosine
  5. Dot product: components
  6. orthogonal vectors
  7. orthogonal projection of u onto a
  8. cross product
• Chapter 4
  1. equality of vectors
  2. norm of a vector (can reuse old card)
  3. Euclidean inner product (can modify components card)
  4. orthogonal vectors (reuse)
  5. domain, codomain, image
  6. transformation
  7. linear transformation
  8. defining properties of linear transformation
  9. one-to-one/injective
  10. Eigenvalue/eigenvector of transformation
• Chapter 5
  1. Vector Space
  2. Subspace
  3. Nullspace
  4. Linear Combination
  5. Span
  6. Linear Independence/Linear Dependence
  7. Basis
  8. Dimension
  9. Row space, Column space
  10. Rank, Nullity
• Chapter 6
  1. Inner Product Space
  2. Orthogonal
  3. Orthogonal Complement, W^{^}
  4. Orthonormal
  5. Orthogonal Projection of u onto W
  6. Component of u orthogonal to W
  7. Transition Matrix
  8. Orthogonal Matrix
• Chapter 7
  1. Eigenvalues
  2. Eigenvector
  3. Eigenspace
  4. Diagonalizable
• Chapter 8
  1. Linear transformation/operator
  2. Composition
  3. Kernel/Nullity
  4. Range/Rank
  5. Onto/Surjective
  6. One-to-One/Injective
  7. Similar
  8. Similarity Invariant
  9. Eigenvalue, eigenvector, eigenspace of a linear transformation
  10. Isomorphism/Bijection

Theorems

Know statements and implications unless otherwise noted – proofs only where noted by *

• Chapter 1
  1. Properties of Matrix Arithmetic *
  2. Properties with Scalar Multiples *
  3. Properties of Zero Matrices *
  4. 1.4.4: Equality of inverses
  5. 1.4.5: Inverse of a product *
  6. Properties of Transpose *
  7. 1.4.10: Inverse of a transpose *
  8. Equivalent Statements
  9. 1.6.3: One-sided inverse is the inverse
  10. More Equivalent statements (up to 6 now)
  11. 1.7.1: Upper/lower triangular theorems (* for first part)
  12. 1.7.2: Symmetric matrices properties *
  13. 1.7.3: invertible/symmetric leads to symmetric inverse *
• Chapter 2
  1. 2.1.2: Formula for inverse of A
  2. 2.1.3: det of triangular matrix
  3. 2.2.2: det of transpose
  4. 2.2.4: det of elementary matrices
  5. 2.3.3: add det(A) to equivalences
  6. 2.3.4: det of product
• Chapter 3
  1. 3.2.1: properties of vector arithmetic *
  2. norm of ku
  3. 3.3.1 norm of v as dot product *
  4. 3.3.2 properties of dot product *
  5. 3.4.2: properties of cross product *
  6. 3.4.1: properties of cross and dot product *
• Chapter 4
  1. 3.2.1, 4.1.1: properties of vector arithmetic * (reuse old card)
  2. 3.3.2, 4.1.2: properties of dot product * (reuse old card)
  3. 4.1.3: cauchy-schwarz inequality
  4. pg 176: matrix formula of dot product
  5. 4.3.3: standard matrix of T
  6. Add one-to-one and onto to list of equivalences
  7. matrix of inverse of transformation is inverse of matrix of transformation *
• Chapter 5
  1. 5.1.1: Properties of vectors *
  2. 5.2.1: Testing for a Subspace
  3. 5.2.2: Nullspace is a subspace *
  4. 5.2.3: Span(S) is a subspace (* for (a))
  5. 5.3.1, 5.3.2: Linear Independence Theorems
  6. 5.4.1: Uniqueness of Basis Representation/Coordinate Vector
  7. 5.4.2, 5.4.3: Theorems leading up to dimension
  8. 5.4.5, 5.4.6, 5.4.7: Consequences to the Plus/Minus Theorem
  9. 5.5.3, 5.5.4, 5.5.5, 5.5.6: Finding Bases for the Row Space, Column Space, and Nullspace
  10. 5.6.3 and Factoid in Notes: The Dimension Theorem and Implications
• Chapter 6
  1. 6.1.1: Properties of Inner Products *
  2. 6.2.1: Cauchy-Schwarz Inequality
  3. 6.2.2: Properties of Length *
  4. 6.2.5: Properties of Orthogonal Complements (* for (a) and (b))
  5. 6.2.6: Null space and row space are orthogonal complements
  6. 6.3.1: Coordinates with respect to an orthonormal basis
  7. 6.3.3: Orthogonal sets are linearly independent *
  8. 6.3.4: Projection Theorem
  9. 6.3.5: Projection onto O.N./orthogonal
  10. 6.3.6: Gram-Schmidt Process
  11. 6.4.4: Least squares solution
  12. 6.6.1-4: Properties of Orthogonal Matrices (* for 6.6.2)
• Chapter 7
  1. 7.1.1: Theory that enabled us to find eigenvalues/vectors
  2. 7.1.2: Eigenvalues of triangular matrices
  3. 7.1.3: Eigenvalues/vectors of A^k *
  4. 7.1.4: Invertible matrices can’t have 0 eigenvalues *
  5. 7.2.1: Diagonalization
  6. 7.2.2: Eigenvectors from distinct eigenvalues are LI
  7. 7.2.3: n eigenvalues implies A is diagonalizable
  8. 7.3.1: Orthogonal diagonalization
  9. 7.3.2: Eigenvalues/vectors of symmetric matrices
• Chapter 8
  1. 8.1.1: Properties of Linear Transformations *
  2. 8.2.1: Kernel/Range are subspaces *
  3. 8.2.3: Dimension Theorem
  4. 8.3.2: Equivalences to 1-1 *
  5. 8.5.2: Change of basis (Transition Matrix)
  6. 8.6.1-3: Isomorphism theorems

Routine Calculations

Be able to perform these calculations quickly/routinely (and know where it is/isn’t possible):

• Chapter 1
  1. Gauss-Jordan elimination
  2. Add/Subt/Mult/Scalar Mult./Transpose/Trace (Matrices)
  3. Find inverse of matrix
  4. Solve system using inverses
• Chapter 2
  1. Calculate determinant by cofactor expansion
  2. Find adjoint/cofactor matrix
  3. Find inverse by adjoint formula
  4. Solve by Cramer's rule
  5. Find determinant by row reduction
  6. Find eigenvalues/eigenvectors
  7. Find number of inversions for permutation (also: even/odd)
  8. Find determinant using combinatorial definition
• Chapter 3
  1. add/subt/scalar mult vectors
  2. Find norm/distance
  3. Compute dot product, find angle between vectors, orthogonal projection
  4. Find cross product
  5. Find equation of a plane
• Chapter 4
  1. add/subt/scalar mult vectors
  2. Find norm
  3. Compute dot product, apply properties to prove identities
  4. Find matrices of linear transformations (of transformation given either by name or by formula, including compositions)
  5. Determine if transformation is one-to-one or onto
  6. Find inverse transformations
  7. Give geometric interpretations of eigenvectors/eigenvalues, Find eigenvalues/eigenvectors
  8. Treat polynomials as vectors, find matrices of transformations
  9. Encode/decode with Hill cypher
• Chapter 5
  1. Determine (and prove/disprove) if a set is vector space
  2. Determine (and prove/disprove) if a subset is subspace
  3. Determine if a vector is in the span of a given set
  4. Find span of a set
  5. Provide spanning set for common spaces
  6. Determine if a set of vectors are linearly independent/dependent
  7. Find a coordinate vector with respect to a given basis
  8. Find a basis for a given space
  9. Find dimension of a space
  10. Find a basis for the span of a given set (either of vectors from that set or not)
  11. Find bases/dimensions for row space, column space, and/or nullspace
• Chapter 6
  1. Determine (and prove/disprove) inner products
  2. Given an inner product, find norm, unit sphere, or angle between vectors
  3. State and apply Cauchy-Schwarz Inequality
  4. Find basis for orthogonal complement of a space
  5. Given an orthogonal/orthonormal basis, express a given vector as a linear combination of the basis vectors or find its coordinate vector
  6. Find orthogonal projections
  7. Given a basis for a space, construct an orthonormal basis
  8. Solve/set-up a least squares regression problem
  9. Find transition matrix from a given basis to a given basis
  10. Change coordinates on a vector under a change of basis
• Chapter 7
  1. Find eigenvalues, eigenvectors, eigenspaces
  2. Diagonalize a matrix
  3. Find powers of a matrix
  4. Orthogonally diagonalize a matrix
• Chapter 8
  1. Determine (prove/disprove) linear transformations
  2. Given a basis and the effect of a linear transformation on that basis, find a formula for the transformation
  3. Find kernel/range of a linear transformation
  4. Determine if a linear transformation is surjective, injective, and/or bijective
  5. Find the matrix for a linear transformation given bases for V and W
  6. Determine (prove/disprove) similarity invariants
  7. Find a basis relative to which the matrix of a transformation is diagonal

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