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Vector Spaces: Subspaces, Linear Independence, Basis & Dimension

Explore properties of vector spaces, including subspaces, linear independence, basis, and dimension. Learn about solution spaces, inconsistent systems, and geometric interpretations. Theorems and examples are provided to illustrate concepts.

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Vector Spaces: Subspaces, Linear Independence, Basis & Dimension

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  1. Chapter 5 General Vector Spaces

  2. 5.1 REAL VECTOR SPACES

  3. In this example we will find it convenient to verify the axioms in the following order: 1, 6, 2, 3, 7, 8, 9, 4, 5, and 10. Let

  4. EXAMPLE 5 A Set That Is Not a Vector Space

  5. The Zero Vector Space

  6. THEOREM 5.1.1

  7. 5.2 SUBSPACES THEOREM 5.2.1

  8. list of subspaces of and

  9. Solution Spaces of Homogeneous Systems If AX=b is a system of linear equations, then each vector x that satisfies this equation is called a solution vector of the system. The following theorem shows that the solution vectors of a homogeneous linear system form a vector space, which we shall call the solution space of the system.

  10. THEOREM 5.2.2

  11. This system of equations is inconsistent

  12. THEOREM 5.2.3

  13. EXAMPLE 10 Spaces Spanned by One or Two Vectors

  14. This system is consistent for all k1 ,k2 , and k3 if and only if the coefficient matrix

  15. THEOREM 5.2.4

  16. 5.3 LINEAR INDEPENDENCE

  17. EXAMPLE 1 A Linearly Dependent Set EXAMPLE 2 A Linearly Dependent Set

  18. EXAMPLE 3 Linearly Independent Sets

  19. EXAMPLE 4 Determining Linear Independence/Dependence

  20. THEOREM 5.3.1

  21. EXAMPLE 6 Example 1 Revisited

  22. THEOREM 5.3.2 (a) A finite set of vectors that contains the zero vector is linearly dependent. (b) A set with exactly two vectors is linearly independent if and only if neither vector is a scalar multiple of the other.

  23. Geometric Interpretation of Linear Independence

  24. THEOREM 5.3.3

  25. 5.4 BASIS AND DIMENSION Nonrectangular Coordinate Systems

  26. THEOREM 5.4.1

  27. Thus S is a basis for R3; it is called the standard basis for R3. Looking at the coefficients of i, j, and k in 1, it follows that the coordinates of v relative to the standard basis are a, b, and c, so

  28. so S is a basis for R3

  29. To see that S is linearly independent, assume that

  30. THEOREM 5.4.2 THEOREM 5.4.3 All bases for a finite-dimensional vector space have the same number of vectors.

  31. DEFINITION The dimension of a finite-dimensional vector space V, denoted by dim(V), is defined to be the number of vectors in a basis for V. In addition, we define the zero vector space to have dimension zero.

  32. EXAMPLE 9 Dimensions of Some Vector Spaces

  33. EXAMPLE 10 Dimension of a Solution Space Determine a basis for and the dimension of the solution space of the homogeneous system

  34. {V1,V2} is a basis, and the solution space is two-dimensional.

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