Friedberg Insel And Spence Linear Algebra Pdf

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Arnold J.

Rohit kumar Maurya. Download PDF. A short summary of this paper. Originally, I had intended the document to be used only by a student who was well-acquainted with linear algebra. However, as the document evolved, I found myself including an increasing number of problems. Therefore, I believe the document should be quite comprehensive once it is complete. I do these problems because I am interested in mathematics and consider this kind of thing to be fun.

If you find any errors regardless of subtlety in the document, or you have different or more elegant ways to approach something, then I urge you to contact me at the e-mail address supplied above. This document was started on July 4, This document is currently a work in progress. The solutions to the exercises from this section are very basic and as such have not been included in this document. Matrices and n-tuples are also introduced.

Some elementary theorems are stated and proved, such as the Cancellation Law for Vector Addition Theorem 1. Prove Corollaries 1 and 2 [uniqueness of additive identities and additive inverses] of Theorem 1. Let V denote the set of all differentiable real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication defined in Example 3.

Therefore, V satisfies VS 1. Therefore, V satisfies VS 2. Therefore V satisfies VS 3. Therefore V satisfies VS 4. So VS 5 is satisfied. This is because multiplication in R is associative. So VS 6 is satisfied. This is because multiplication in R is distributive over addition. So VS 7 is satisfied.

This is again because multiplication in R is distributive over addition. So VS 8 is satisfied. This completes the proof that V, together with addition and scalar multiplication as defined in Example 3, is a vector space.

There are 2mn vectors in this vector space. A certain theorem that is referred to as the subspace test is stated and proven, which provides a fast way of checking whether a given subset is a subspace. More concepts relating to matrices are introduced as well.

The section closes with a proof that the intersection of two subspaces is itself a subspace. Let W1 and W2 be subspaces of a vector space V. Prove that if W is a subspace of a vector space V and w1 , w2 ,.

Since W1 , W2 are subspaces, they both contain the additive identity 0 from V. The definitions of W1 and W2 tell us that if a1 , a2 ,. This clearly only holds for the n- tuple 0, 0,. Let W be a subspace of a vector space V over a field F.

First, we prove the well-definedness of addition. Second, we prove the well-definedness of scalar multiplication. The span of a subset is introduced, in addition to the notion of a generating subset. Some geometric interpretations of the concepts are given. Solve the following systems of linear equations by the method introduced in this section. Begin by replacing equation 1. Divide equation 1. Now replace equation 1. Switch equations 1.

This solution set represents a plane in R3 because there are two parameters s and t. Next, replace equation 1. Replace equation 1. Now, replace equation 1. For each of the following lists of vectors in R3 , determine whether the first vector can be expressed as a linear combination of the other two.

For each list of polynomials in P3 R , determine whether the first polynomial can be expressed as a linear combination of the other two. Interpret this result geometrically in R3. In R3 this result tells us that the span of any single vector will be a line passing through the origin. This proves that W is closed under addition. This proves that W is closed under scalar multiplication. By the subspace test, we have that W is a subspace.

Then, by the definition of span W , we see that a is a linear combination of vectors in W. Then for a1 , a2 ,. However, each element in S2 is also in V. Since V is closed under vector addition and scalar multiplication, surely every linear combination of vectors in S2 must be in V. The sum of two subsets is defined in the exercises of Section 1. There- fore, we can write, for some a1 , a2 ,. Since by Theorem 1. Then we can write, for some a1 , a2 ,. Let S1 and S2 be subsets of a vector space V.

Let V be a vector space and S a subset of V with the property that whenever v1 , v2 ,. Prove that every vector in the span of S can be uniquely written as a linear combination of vectors of S. This clearly proves that our two representations of v cannot be distinct, as required. Let W be a subspace of a vector space V. Under what conditions are there only a finite number of distinct subsets S of W such that S generates W? Afterwards, some proofs concerning linear dependence and linear independence are given.

Determine whether the following sets are linearly dependent or linearly independent. In Fn , let ej denote the vector whose jth coordinate is 1 and whose other coordinates are 0. Recall from Example 3 in Section 1. Find a linearly independent subset that generates this subspace. Let u and v be distinct vectors in a vector space V. This completes the proof. How many vectors are there in span S?

Justify your answer. The span of S will consist of every linear combination of vectors in S, or in some subset of S. The field Z2 only admits two possible scalars, 0 and 1, therefore every subset of S can be regarded as a linear combination and vice versa. The power set of S that is, the set of all subsets of S will contain 2n elements by a theorem in set theory , since S contains n elements.

There is therefore a bijection between the set of all subsets of S and the set of all linear combinations of vectors in S. For this reason, the cardinality of span S must also be 2n. Prove Theorem 1. Assume S1 is linearly dependent. We wish to show that S2 is linearly dependent. Since S1 is linearly dependent, there exist a1 , a2 ,. For this reason, S2 is also linearly dependent. Assume S2 is linearly independent.

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