Linear

Handbook of linear algebra by A. R. Heesterman

By A. R. Heesterman

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Example text

8. 9. 10. Span(∅) = {0}. A linear combination of a single vector v is simply a scalar multiple of v. In a vector space V , Span(v1 , v2 , . . , vk ) is a subspace of V . Suppose the set of vectors S = {v1 , v2 , . . , vk } spans the vector space V . If one of the vectors, say vi , is a linear combination of the remaining vectors, then the set formed from S by removing vi still spans V . Any single nonzero vector is linearly independent. Two nonzero vectors are linearly independent if and only if neither is a scalar multiple of the other.

Linear Transformations Francesco Barioli . . . . . . . . . . . . . . . . . . . Determinants and Eigenvalues Luz M. DeAlba . . . . . . . . . . . . . . . . Inner Product Spaces, Orthogonal Projection, Least Squares, and Singular Value Decomposition Lixing Han and Michael Neumann . . . . . 1-1 2-1 3-1 4-1 5-1 Matrices with Special Properties 6 7 8 9 10 Canonical Forms Leslie Hogben . . . . . . . . . . . . . . .

5. The n vectors e1 = ⎢ 0 ⎥ , e2 = ⎢ 0 ⎥ , . . , en = ⎢ .. ⎥ span F n , for any field F . ⎦ ⎣0⎦ 0 0 1 also linearly independent. 6. In R2 , 1 −1 and dependent, because 0 3 are linearly independent. However, 1 5 = 1 , −1 0 , and 3 1 5 are linearly 1 0 +2 . −1 3 7. The infinite set {1, x, x 2 , . . , x n , . } is linearly independent in F [x], for any field F . 8. In the vector space of continuous real-valued functions on the real line, C(R), the set {sin(x), sin(2x), . . , sin(nx), cos(x), cos(2x), .