Linear Algebra
... We call a vector space finite dimensional iff we can find a basis containing only a finite number of elements. Otherwise, we call it infinite dimensional. The following so called basis theorem is fundamental. Theorem 5.2. The number of elements in any basis of a vector space X is the same as in any ...
... We call a vector space finite dimensional iff we can find a basis containing only a finite number of elements. Otherwise, we call it infinite dimensional. The following so called basis theorem is fundamental. Theorem 5.2. The number of elements in any basis of a vector space X is the same as in any ...
Vector Spaces 1 Definition of vector spaces
... of vectors and scalar multiplication. These operations satisfy certain properties, which we are about to discuss in more detail. The scalars are taken from a field F, where for the remainder of these notes F stands either for the real numbers R or the complex numbers C. The real and complex numbers ...
... of vectors and scalar multiplication. These operations satisfy certain properties, which we are about to discuss in more detail. The scalars are taken from a field F, where for the remainder of these notes F stands either for the real numbers R or the complex numbers C. The real and complex numbers ...
Chapter 3
... (ii) AX = 0n×1 has only the trivial solution X = 0n×1 ; (iii) the reduced row echelon form of A is In ; (iv) A is row equivalent to In ; (v) A can be written as a product of elementary matrices. Proof. We prove (i) =⇒ (ii) =⇒ (iii) =⇒ (iv) =⇒ (v) =⇒ (i). (i) =⇒ (ii). Suppose A is invertible. If AX = ...
... (ii) AX = 0n×1 has only the trivial solution X = 0n×1 ; (iii) the reduced row echelon form of A is In ; (iv) A is row equivalent to In ; (v) A can be written as a product of elementary matrices. Proof. We prove (i) =⇒ (ii) =⇒ (iii) =⇒ (iv) =⇒ (v) =⇒ (i). (i) =⇒ (ii). Suppose A is invertible. If AX = ...
