Chapter 6 General Linear Transformations
... General Linear Transformations 6.1 Linear Transformations from Rn to Rm We studied linear transformations in Section 4.2 and used the following definition to determine whether a transformation from Rn to Rm is linear. Definition 6.1.1. A transformation T : Rn → Rm is linear if the following two prop ...
... General Linear Transformations 6.1 Linear Transformations from Rn to Rm We studied linear transformations in Section 4.2 and used the following definition to determine whether a transformation from Rn to Rm is linear. Definition 6.1.1. A transformation T : Rn → Rm is linear if the following two prop ...
basic matrix operations
... This is a 2 3 matrix. A matrix with m rows and n columns has dimensions or size m n . The number of rows is always given first. A matrix with only one row is called a row matrix or row vector. A matrix with only one column is called a column matrix or column vector. A matrix with the same number o ...
... This is a 2 3 matrix. A matrix with m rows and n columns has dimensions or size m n . The number of rows is always given first. A matrix with only one row is called a row matrix or row vector. A matrix with only one column is called a column matrix or column vector. A matrix with the same number o ...
Homework 1. Solutions 1 a) Let x 2 + y2 = R2 be a circle in E2. Write
... b) det G = A(u, v)D(u, v) − B(u, v)C(u, v) = AD − B 2 6= 0 since it is non-degenerate (see the solution of exercise 1) c) Consider quadratic form G(x, x) = gik xi xk = Ax2 +2Bxy+Dy 2 . (We already know that B = C) Positive -definiteness means that G(x, x) > 0 for all x 6= 0. In particular if we put ...
... b) det G = A(u, v)D(u, v) − B(u, v)C(u, v) = AD − B 2 6= 0 since it is non-degenerate (see the solution of exercise 1) c) Consider quadratic form G(x, x) = gik xi xk = Ax2 +2Bxy+Dy 2 . (We already know that B = C) Positive -definiteness means that G(x, x) > 0 for all x 6= 0. In particular if we put ...
Linear Transformations
... Theorem: A linear transformation T : V → W is one-to-one if and only if ker(T ) = {~0}. Theorem: Let T : V → V be a linear operator, where V is a finite dimensional vector space. The following statements are equivalent. a) T is one-to-one b) ker(T ) = {~0} c) T is onto. Definition: A linear transfor ...
... Theorem: A linear transformation T : V → W is one-to-one if and only if ker(T ) = {~0}. Theorem: Let T : V → V be a linear operator, where V is a finite dimensional vector space. The following statements are equivalent. a) T is one-to-one b) ker(T ) = {~0} c) T is onto. Definition: A linear transfor ...
Review
... vector space V, then W is a subspace of V if and only if 1. If u and v are in W, then u+v is in W. 2. If u is in W and c is any scalar, then cu is in W. ...
... vector space V, then W is a subspace of V if and only if 1. If u and v are in W, then u+v is in W. 2. If u is in W and c is any scalar, then cu is in W. ...
Week_2_LinearAlgebra..
... • If the spanning set is linearly independent, it’s also known as a basis for that subspace • The coordinate representation of a vector in a subspace is unique with respect to a basis for that subspace ...
... • If the spanning set is linearly independent, it’s also known as a basis for that subspace • The coordinate representation of a vector in a subspace is unique with respect to a basis for that subspace ...
