Fundamental Theorem of Linear Algebra

We will need the Fundamental Theorem of Linear Algebra in Section 7.1. 

Proposition 2.5 (Fundamental Theorem of Linear Algebra) For any linear transformation T with finite-dimensional domain V we have... 

In Section 7.1 we will use this characterization of homogeneous harmonic polynomials as a kernel of a linear transformation (along with the Fundamental Theorem of Linear Algebra, Proposition 2.5) to calculate the dimensions of the spaces of the spherical harmonics. 

To calculate the dimension of the vector space H for every nonnegative integer f we will use the Fundamental Theorem of Linear Algebra (Proposition 2.5), which we repeat here if T is a linear transformation from a finitedimensional vector space V to a finite-dimensional vector space W, then we have... 

Our goal is to calculate the dimension of the kernel of V, since this kernel consists precisely of the harmonic functions in. From Section 2.2 we know that the dimension of P is (f -I- l)(f -f 2). So, by the Fundamental Theorem of Linear Algebra (Proposition 2.5) it suffices to calculate the dimension of the image of the the linear transformation V. ... 

Exercise 2.9 Null space and fundamental theorem of linear algebra... 

Equation 2.67 in Exercise 2.8 is begging to be analyzed by the fundamental theorem of linear algebra [7], so we explore that concept here. Consider an arbitrary m x n matrix, B. The null space of matrix B, written 3 B), is defined to be the set of all vectors x such that S x = 0. The dimension of SEiB) is the. number of linearly independent vectors x satisfying B x == 0. One of the remarkable results of the fundamental theorem of linear algebra is the relation... 

This proof does not give a method for finding real eigenvalues of real linear operators, because the Fundamental Theorem of Algebra does not guarantee real roots for polynomials with real coefficients. Proposition 2.11 does not hold for inhnite-dimensional complex vector spaces eiffier. See Fxercise 2.28. 

Similarly, the number of linearly independent rows of A is called the row-rank of A. The row-rank of A is the column-rank of A. A fundamental theorem in matrix algebra states that the row-rank and the column-rank of a matrix are equal (and equal to the rank) [Schott 1997], Hence, it follows that the rank r(A) < min(/,/). The matrix A has full rank if and only if r(A) = mini/,/). Sometimes the term full column-rank is used. This means that r(A) = min(/,/) =. /, implying that J < I. The term full row-rank is defined analogously. 

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