MATH 51 MIDTERM 1 SOLUTIONS 1. Compute the following: (a). 1
... (a). Suppose that a linear subspace V is spanned by vectors v1 , v2 , . . . , vk . What, if anything, can you conclude about the dimension of V ? Answer: dimV ≤ k. Since we have k vectors spanning V a set with more than k vectors must be linearly dependent (proposition 12.1 in the book). Since a bas ...
... (a). Suppose that a linear subspace V is spanned by vectors v1 , v2 , . . . , vk . What, if anything, can you conclude about the dimension of V ? Answer: dimV ≤ k. Since we have k vectors spanning V a set with more than k vectors must be linearly dependent (proposition 12.1 in the book). Since a bas ...
Analysis on arithmetic quotients Chapter I. The geometry of SL(2)
... one is dealing with P . The relationship expressed by the Cayley transform is extremely important even when dealing with more general Lie groups. It is used to keep track of the collection of conjugacy classes of algebraic tori—in effect, it conjugates the compact torus of SL2 (R) to the split torus ...
... one is dealing with P . The relationship expressed by the Cayley transform is extremely important even when dealing with more general Lie groups. It is used to keep track of the collection of conjugacy classes of algebraic tori—in effect, it conjugates the compact torus of SL2 (R) to the split torus ...
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9. Change of basis/coordinates Theorem Let β and β be two ordered
... Let β and β 0 be two ordered bases for a finite dimensional vector space V . We have: ...
... Let β and β 0 be two ordered bases for a finite dimensional vector space V . We have: ...
xi. linear algebra
... sigma indicates the sum. We can make a few observations (indirectly) from these properties. First, a square matrix A is invertible if and only if zero is not an eigenvalue of A. Second, if ! is an eigenvalue of A and A is invertible, then ! "1 is an eigenvalue of A !1 . Third, if A and B are both n ...
... sigma indicates the sum. We can make a few observations (indirectly) from these properties. First, a square matrix A is invertible if and only if zero is not an eigenvalue of A. Second, if ! is an eigenvalue of A and A is invertible, then ! "1 is an eigenvalue of A !1 . Third, if A and B are both n ...
Orthogonal Polynomials
... Here is the analogy to the case of the least-squares technique over a vector space. In the space of all functions, the orthogonal polynomials p0 , . . . pk constitute an “orthogonal basis” for the subspace of polynomial functions of degree no more than k. The least-squares approximation of a functio ...
... Here is the analogy to the case of the least-squares technique over a vector space. In the space of all functions, the orthogonal polynomials p0 , . . . pk constitute an “orthogonal basis” for the subspace of polynomial functions of degree no more than k. The least-squares approximation of a functio ...
Section 1
... letters. Given a length unit, and two orthogonal lines of reference called the x-axis and the y-axis, each point P ∈ E2 can be represented by an ordered pair of real numbers (x, y) measuring the perpendicular distance of P from the y- and x-axis, respectively. The axes intersect at the origin O, wit ...
... letters. Given a length unit, and two orthogonal lines of reference called the x-axis and the y-axis, each point P ∈ E2 can be represented by an ordered pair of real numbers (x, y) measuring the perpendicular distance of P from the y- and x-axis, respectively. The axes intersect at the origin O, wit ...
Linear Algebra - John Abbott Home Page
... Science Program. It is generally taken in the third semester. Linear Algebra introduces the student to matrices and vector spaces with applications to Business, Commerce and Computer Science. The primary purpose of the course is the attainment of Objectives 022Z, 022R, 022S (“To apply methods of lin ...
... Science Program. It is generally taken in the third semester. Linear Algebra introduces the student to matrices and vector spaces with applications to Business, Commerce and Computer Science. The primary purpose of the course is the attainment of Objectives 022Z, 022R, 022S (“To apply methods of lin ...
Lecture 33 - Math TAMU
... Definition. Let V be a vector space and L : V → V be a linear operator. A number λ is called an eigenvalue of the operator L if L(v) = λv for a nonzero vector v ∈ V . The vector v is called an eigenvector of L associated with the eigenvalue λ. (If V is a functional space then eigenvectors are also c ...
... Definition. Let V be a vector space and L : V → V be a linear operator. A number λ is called an eigenvalue of the operator L if L(v) = λv for a nonzero vector v ∈ V . The vector v is called an eigenvector of L associated with the eigenvalue λ. (If V is a functional space then eigenvectors are also c ...
570 SOME PROPERTIES OF THE DISCRIMINANT MATRICES OF A
... (i = 1, 2). Since £i is idempotent and Fis non-modular, ti(e\) =^0, (i = 1, 2). Hence by a transformation of basis of A which leaves the basis of N unchanged and the principal unit of B invariant, Ti(A) [or 7^2C4)] can be reduced to a diagonal form ||gr8r*||, where g* = 0, (i>p). A manipulation of t ...
... (i = 1, 2). Since £i is idempotent and Fis non-modular, ti(e\) =^0, (i = 1, 2). Hence by a transformation of basis of A which leaves the basis of N unchanged and the principal unit of B invariant, Ti(A) [or 7^2C4)] can be reduced to a diagonal form ||gr8r*||, where g* = 0, (i>p). A manipulation of t ...
Linear Algebra and Matrices
... linear combination of the others. They define in space a smaller number of dimensions than the total number of vectors in the set. The resulting matrix will be rank-deficient and the determinant will be zero. Similarly, if all the elements of a line or column are zero, the determinant of the matrix ...
... linear combination of the others. They define in space a smaller number of dimensions than the total number of vectors in the set. The resulting matrix will be rank-deficient and the determinant will be zero. Similarly, if all the elements of a line or column are zero, the determinant of the matrix ...