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# Linear Algebra

## Prove that, every basis of a vector space V has the same number of elements.

Problem: Prove that, every basis of a vector space V has the same number of elements.  Proof: Suppose {u1 , u2 , . . . , un } is a basis of  V, and suppose {v1 , v2 , . . .} is another basis of V. Because {ui} spans ...

## Cayley- Hamilton Theorem: Every Matrix A is a root of its characterstic polynomial ∆(t).

Cayley- Hamilton Theorem: Every Matrix A is a root of its characterstic polynomial ∆(t). Proof: Let A be an arbitrary n- square matrix and let ∆(t) be its characterstic polynomial, say, ∆(t) = |tI-A| = tn +an-1tn-1+ …+a1t+a0 Now let B(t) denote the classical adjoint of the matrix tI-A. The ...

## Linear Mapping| Linear Algebra

Mapping and Functions: Let A and B be arbitrary nonempty sets. Suppose to each element in A there is assigned a unique element of B; the collection f of such assignment is called a mapping ( or map) from A into B, and is denoted by f:A⟶B. The set A ...

## Linear combination, Spanning set| Linear Algebra

Linear combination: Let V be a vector space over a field K. A vector v in V is a linear combination of vectors u1, u2,—–, um in V if there exist scalars a1, a2, —–, am in K such that v= a1u1+ a2u2+—+ amum Example: (Linear combination in Rn): Suppose ...

## Elementary matrices| Linear Algebra

Elementary matrices: Let e denote an elementary row operation and let e(A) denote the result of applying the operation e to matrix. Now let E be the matrix obtained by applying to the identity matrix I, that is, E = e(I) Then E is called the elementary matrix corresponding to ...

## Vector Spaces| Chapter 4|linear algebra

Definition: Let V be a nonempty set of vectors with two operations: (i) Vector addition: This assigns to any u,v ∊V, k∊K a sum u+v in V. (ii) Scalar multiplication: This assigns to any u ∊V, k ∊K a product ku ∊V. Then V is called a vector space if ...

## Echelon matrices, Row canonical form, Row equivalence

Echelon matrices: A matrix A is called an echelon matrix, or is said to be in echelon form, if the following two conditions: (1) All zero rows, if any, are at the bottom of the matrix. (2) Each leading non-zero entry in a row is to the right of leading ...

## Basis and Dimension

Definition of basis: (A) A basis for a vector space is a sequence of vectors that is linearly Independent and that spans the space. (B) A set S={u1 , u2 , ….. , un}of vectors is a basis of V if it has the following two properties: (i) S is ...

## Linear equations

Definition: A linear equation in unknowns x1, x2,—————xn is an equation that can be put in the standard form a1 x1+ a2 x2+ —————+ an xn = b where a1, a2,—————an and b are constants. The constant ak is called the coefficient of xk, and b is called the constant ...

## Algebra of matrice

Definition: A matrix A over a field K or, simply, a matrix A is a regular array of scalars usually presented in the following form The rows of such a matrix A are the m horizontal lists of scalars: and the terms of A are the n verticals lists of ...