Chapter |
You need to know... ("Def" means definition...) |
1 |
- Def of linear combination
- Dot product
- Length of and angles between vectors.
- Compute the distance between a point and a line in R2.
- Understand and use the parametric equations of lines, planes
and subspaces in Rn.
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2 |
- Def of a linear equations
- Find the solution of a system of linear equations
- Find the row-echelon form of a matrix.
- Apply elementary row operations to a matrix.
- Def of linear independence of a set of vectors.
- State linear independence in different ways.
- Def of the span of a set of vectors
- How many solutions can a linear system have? (Hint: There are
three possibilities)
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3 |
- Compute the product of two matrices
- Def of invertible matrix
- Find the inverse of a matrix, if possible, applying two
different methods (Gauss-Jordan elimination and using
determinants, as done in Chapter 4)
- Solve a linear system of equations using the inverse of a
matrix
- Def. of elementary matrix.
- Def. of subspace
- Def of basis of a subspace
- Def of dimension of a subspace
- Def of the row and column space of a matrix
- Def of nullspace of a matrix
- Def of rank of a matrix
- Find the coordinates of a vectors with respect to a given
basis.
- Def of linear transformation and relation to matrices
- Composition of linear transformation and relation product of
matrices
- Inverse of a linear transformation (when possible) and
relation to inverse of the corresponding matrix.
- Find the matrix of a rotation in R2
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4 |
- Compute the determinant of a matrix
- Properties of the determinant.
- Relation between the inverse of an invertible matrix and its
determinant.
- Find the characteristic polynomial of a square matrix.
- Definition of eigenvalues, eigenvectors and eigenspaces.
- Find eigenvalues, eigenvectors and eigenspaces of a given
matrix.
- Compute the power of a matrix using eigenvalues and
eigenvectors.
- Def. of diagonalizable matrix
- For a diagonalizable matrix, find an invertible matrix P and a
diagonal matrix D such that P-1AP=D
- Def of geometric and algebraic multiplicity of an eigenvalue
- Relation between geometric and algebraic multiplicity of an
eigenvalue.
- Find the matrix of a reflection about a line in R2.
- Find the matrix of a reflection about a plane in R3.
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5 |
- Def of orthonormal set.
- Def of orthogonal set.
- Relation between orthogonality and linear intependence
- Write a vector in a subspace W as a linear combination of
elements of an orthonormal basis of W
- Write a vector in a subspace W as a linear combination of
elements of an orthogonal basis of W
- Def of orthogonal matrix
- Inverse of an orthogonal matrix
- Eigenvalues of an orthogonal matrix.
- Def. of orthogonal complement of a subspace
- Dimension of the orthogonal complement of a subspace
- Find the orthogonal complement of a subspace.
- Compute the orthogonal projection of a vector onto a subspace
- Decompose a vector into an element of a subspace W and one
element of the orthogonal complement of W
- Given a basis of a subspace W, compute an orthonormal basis
applying the Gram-Schmidt process.
- Given a symmetric matrix A, find an orthogonal matrix Q and a
diagonal matrix D such that D=QTAQ
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