By Giuliano Sorani

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**Extra resources for An introduction to real and complex manifolds.**

**Example text**

Hence AB as just deﬁned is an m × p matrix. 6 Multiply the following. ( 1 2 0 2 1 1 ) 1 2 0 0 3 1 −2 1 1 The ﬁrst thing you need to check before doing anything else is whether it is possible to do the multiplication. The ﬁrst matrix is a 2 × 3 and the second matrix is a 3 × 3. Therefore, is it possible to multiply these matrices. According to the above discussion it should be a 2 × 3 matrix of the form First column Second column Third column � �� � �� � �� � � � ( ) ( ) ( ) 1 2 0 1 2 1 1 2 1 1 2 1 0 , 3 , 1 0 2 1 0 2 1 0 2 1 −2 1 1 You know how to multiply a matrix times a three columns.

There is one way to go from location 1 to location 3. com 55 Linear Algebra I Matrices and Row operations Linear Transformations from location 2 to location 3 although it is possible to go from location 3 to location 2. Lets refer to moving along one of these directed lines as a step. The following 3 × 3 matrix is a numerical way of writing the above graph. This is sometimes called a digraph, short for directed graph. 1 1 1 1 0 0 1 1 0 Thus aij , the entry in the ith row and j th column represents the number of ways to go from location i to location j in one step.

Prove from the axioms of the inner product the parallelogram identity, |a + b| + 2 2 2 |a − b| = 2 |a| + 2 |b| . ∑n 3. For a, b ∈ Rn , deﬁne a · b ≡ k=1 β k ak bk where β k > 0 for each k. Show this satisﬁes the axioms of the inner product. What does the Cauchy Schwarz inequality say in this case. 4. In Problem 3 above, suppose you only know β k ≥ 0. Does the Cauchy Schwarz inequality still hold? If so, prove it. 5. Let f, g be continuous functions and deﬁne ∫ 1 f ·g ≡ f (t) g (t)dt 0 show this satisﬁes the axioms of a inner product if you think of continuous functions in the place of a vector in Fn .