This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 311, Homework 8 partial answers and solutions 3.5C.1. (a) Any matrix A with entries a ij is a linear combination of the matrices E ij : A = m X i =1 n X j =1 a ij E ij , so these matrices span all of M m,n . Conversely, any linear combination m X i =1 n X j =1 a ij E ij is the matrix with entries a ij . So if this linear combination is the zero matrix, then all its entries, and so all the coefficients in the linear combination, have to be zero. This means that the matrices { E ij } are linearly independent. As a result, they form a basis, and the dimension of the space of m n matrices is mn . (b) A similar argument shows that a basis for the space of diagonal n n matrices are the matrices { E ii : i = 1 , 2 , . . . , n } . As a result, this space has dimension n . 3.5C.3. Suppose a 1 x 1 + a 2 x 2 + . . . + a k x k = . Then using the linearity of the function f , a 1 f ( x 1 ) + a 2 f ( x 2 ) + . . . + a k f ( x k ) = f ( a 1 x 1 + a 2 x 2 + . . . + a k x k ) = f ( ) = ....
View Full
Document
 Spring '08
 Anshelvich
 Math, Matrices

Click to edit the document details