Applied Linear Algebra and Matrix Analysis by Thomas S. Shores

Author:Thomas S. Shores
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

The Dimension Theorem

No doubt you have already noticed that every basis of the vector space must have exactly two elements in it. Similarly, one can reason geometrically that any basis of must consist of exactly three elements. These numbers somehow measure the “size” of the space in terms of the degrees of freedom (number of coordinates) one needs to describe a general vector in the space. The dimension theorem asserts that this number can be unambiguously defined. As a matter of fact, the discussion on Page 214 shows that every basis of has exactly n elements. Our next stop: arbitrary finite-dimensional vector spaces. Along the way, we need a very handy theorem that is sometimes called the Steinitz substitution principle. This principle is a mouthful to swallow, so we will precede its statement with an example that illustrates its basic idea.

Example 3.35.

Let , , , , and . Then form a linearly independent set and form a basis of (assume this). Show how to substitute both and into the set while substituting out some of the ’s and at the same time retaining the basis property of the set.

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