CHANGE OF BASIS MATRIX PDF



Change Of Basis Matrix Pdf

Change of Basis HMC Calculus Tutorial. Theorem. A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. i.e. P 1 = PT: Example Consider R3 with the orthonormal basis S= 8 >> < >>: u 1 = 0 B B @ p2 6 p 1 6 p 1 6 1 C C A;u 2 = 0 B B @ 0 p 2 p 2 1 C C A;u 3 = 0 B B @ 1 3 p 3 p 3 1 C C A 9 >> = >>;: Let Rbe the standard basis fe 1;e 2;e 3g. Since we are changing from the standard basis to a new basis, then, To gain a good undertanding of a self linear transformation, we find a basis (a suitable invertible matrix P) so that the the transformation behaves particularly nice with respect to the basis (the similarity change via P gives a diagonal matrix)..

Worksheet 19 Change of basis MIT Mathematics

5 Vectorspacecoordinateswithrespecttoabasis. Change of. Proof (1 of 2) I Let V is a п¬Ѓnite-dimensional inner product space, let Band B0be two orthonormal bases for V, and let P be the transition matrix from B0to B., Let the columns of matrix W be the basis vectors of the new basis. Then if x is a vector in the old basis, we can convert it to a vector c in the new basis using.

Change of Coordinates with Orthonormal Bases (pages 325-329) Because it is so easy to nd the coordinates with respect to an orthonormal basis B= f~v 1;:::;~v ng, we will easily be able to nd the change of coordinates matrix from the standard basis Sto B. In fact, it will be easier than we could have hoped! After all, the change of coordinates matrix Q from Sto Bsatis es the following: Q = [~e Given coordinates of a point in one basis, we will be able to obtain the coordinates of that point in another basis by applying the change of coordinates matrix to it. В§ A3 .2 Change of Basis Basics Let us first recall a few basic facts about bases and change of basis matrices.

Conversely, if V is any vector space and if v1,...,v n is a basis for V, then we define the matrix [H] v 1,...,v n for H with respect to this basis to be the matrix Final Exam Linear Algebra, Dave Bayer, December 16, 1999 Please work only one problem per page, starting with the pages provided, and number all continuations clearly.Only work which can be found in this way will be graded.

Change of Basis Let $V$ be a vector space and let $S = \{{\bf v_1,v_2, \ldots, v_n}\}$ be a set of vectors in $V$. Recall that $S$ forms a basis for $V$ if the The goal of these notes is to provide an apparatus for dealing with change of basis in vector spaces, matrices of linear transformations, and how the matrix changes when the basis changes.

Chapter & Page: 5–2 Change of Basis and AA† = 3 5 4 5 i 4 5 i 3 5 3 5 −4 5 i −4 5 i 3 5 = ··· = I . So A is unitary. Obviously, for a matrix to be unitary, it must be square. Alternatively, we can find Q ¼ P*1 by forming the matrix M ¼½P;I& and row reducing M to row canonical form: M ¼ 1211 00 0120 10 1220 01 2 6 4 3 7 5+ 10 0*2 *23

• A basis is a linearly independent set of vectors whose combinations will get you anywhere within a space, i.e. span the space • n vectors are required to span an n-dimensional space • If the basis vectors are normalized and mutually orthogonal the basis is orthonormal • There are lots of possible bases for a given vector space; there’s nothing special about a particular basis—but A) Find the change of basis matrix for converting from the standard basis to the basis B. I have never done anything like this and the only examples I can find online basically tell me how to do the change of basis for "change-of-coordinates matrix from B to C".

3 In this case, the Change of Basis Theorem says that the matrix representation for the linear transformation is given by P 1AP. We can summarize this as follows. Change of Basis Let $V$ be a vector space and let $S = \{{\bf v_1,v_2, \ldots, v_n}\}$ be a set of vectors in $V$. Recall that $S$ forms a basis for $V$ if the

3 In this case, the Change of Basis Theorem says that the matrix representation for the linear transformation is given by P 1AP. We can summarize this as follows. Complete Change of basis matrix - Mathematics, Engineering chapter (including extra questions, long questions, short questions) can be found on EduRev, you can check out Engineering Mathematics lecture & lessons summary in the same course for Engineering Mathematics Syllabus. EduRev is like a wikipedia just for education and the Change of basis matrix - Mathematics, Engineering images and

Definition. A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C) is a linearly independent subset of V that spans V. Math 28S Coordinate changes Fall 2011 1 Change of basis matrices Suppose V is a vector space over eld F with dimension n. Then, we know that V ˘=Fn

an invertible matrix P(the associated basis change matrix) such that Pв€’1APis a simple as possible. Our preferred form of matrix is a diagonal matrix, but we saw i n MA106 that the matrix Change of Basis Let $V$ be a vector space and let $S = \{{\bf v_1,v_2, \ldots, v_n}\}$ be a set of vectors in $V$. Recall that $S$ forms a basis for $V$ if the

7.1 Coordinatization and Change of Basis Chapter 7. Change

change of basis matrix pdf

linear algebra Change of basis matrix to convert. 27/12/2015 · The basis are denoted A and B. Similar to the previous video, we again compute a "change-of-coordinates" matrix that can transform a vector written in each basis …, Changing basis changes the matrix of a linear transformation. However, as a map between vector spaces, \(\textit{the linear transformation is the same no matter which basis we use}\)..

Matrix representation and change of basis the special

change of basis matrix pdf

Worksheet 19 Change of basis MIT Mathematics. Proof (1 of 2) I Let V is a п¬Ѓnite-dimensional inner product space, let Band B0be two orthonormal bases for V, and let P be the transition matrix from B0to B. an invertible matrix P(the associated basis change matrix) such that Pв€’1APis a simple as possible. Our preferred form of matrix is a diagonal matrix, but we saw i n MA106 that the matrix.

change of basis matrix pdf


Change of coordinates Math 130 Linear Algebra D Joyce, Fall 2015 The coordinates of a vector v in a vector space V with respect to a basis = fb 1;b Given coordinates of a point in one basis, we will be able to obtain the coordinates of that point in another basis by applying the change of coordinates matrix to it. В§ A3 .2 Change of Basis Basics Let us first recall a few basic facts about bases and change of basis matrices.

is the change of basis matrix of the basis. In order to understand this relationship In order to understand this relationship better, it is convenient to take it as a de nition and then study it abstractly. an invertible matrix P(the associated basis change matrix) such that Pв€’1APis a simple as possible. Our preferred form of matrix is a diagonal matrix, but we saw i n MA106 that the matrix

n form a basis set in a vector space if, and only if, each vector in the space can be represented uniquely as a sum of scalar multiples of the basis vectors, as in (2). There are two requirements Change of basis • Let B denote a matrix whose columns form an orthonormal basis for a vector space W If B is full rank (n x n), then we can get back to the original basis through multiplication by B. Change of basis • Let B denote a matrix whose columns form an orthonormal basis for a vector space W Vector of projections of v along each basis vector . Orthogonal matrix • In this case

In particular it does not change the fact that given the coordinates of a new basis, one needs to compute the inverse of a matrix in order to transform coordinates to that new basis. The two issues are quite unrelated, so changing conventions on one issue does not solve anything for the other issue. 27/12/2015 · The basis are denoted A and B. Similar to the previous video, we again compute a "change-of-coordinates" matrix that can transform a vector written in each basis …

Change of Bases Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 8, 2007) We have seen in previous lectures that linear operators on an n dimensional vector space are in one-to-one correspondence with n×n matrices. This correspondence depends on the choice of basis for the vector space, however. In this lecture we address the question how the matrix for a linear operator changes … • A basis is a linearly independent set of vectors whose combinations will get you anywhere within a space, i.e. span the space • n vectors are required to span an n-dimensional space • If the basis vectors are normalized and mutually orthogonal the basis is orthonormal • There are lots of possible bases for a given vector space; there’s nothing special about a particular basis—but

basis and writing it in terms of the second basis, we would multiply by the inverse of the basis change matrix: 1 Note this istrue forcomposition oflineartransformations forany sizevectorspaces, A matrix A is a linear transformation; to write it in matrix form requires us to choose a coordinate system (basis), and the transformation will look different in different bases.

Change of Coordinates with Orthonormal Bases (pages 325-329) Because it is so easy to nd the coordinates with respect to an orthonormal basis B= f~v 1;:::;~v ng, we will easily be able to nd the change of coordinates matrix from the standard basis Sto B. In fact, it will be easier than we could have hoped! After all, the change of coordinates matrix Q from Sto Bsatis es the following: Q = [~e Similarly, the change-of-basis matrix can be used to show that eigenvectors obtained from one matrix representation will be precisely those obtained from any other representation. So we can determine the eigenvalues and eigenvectors of a linear transformation by forming one matrix representation, using any basis we please, and analyzing the matrix in the manner of Chapter E .

Theorem. A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. i.e. P 1 = PT: Example Consider R3 with the orthonormal basis S= 8 >> < >>: u 1 = 0 B B @ p2 6 p 1 6 p 1 6 1 C C A;u 2 = 0 B B @ 0 p 2 p 2 1 C C A;u 3 = 0 B B @ 1 3 p 3 p 3 1 C C A 9 >> = >>;: Let Rbe the standard basis fe 1;e 2;e 3g. Since we are changing from the standard basis to a new basis, then Change of coordinates Math 130 Linear Algebra D Joyce, Fall 2015 The coordinates of a vector v in a vector space V with respect to a basis = fb 1;b

Change of Basis HMC Calculus Tutorial

change of basis matrix pdf

Change of Basis 2 Matrices Department of Mathematics. Assignment 4/MATH 247/Winter 2010 . Due: Tuesday, February 9 . First, you find, in Part 1, a summary of some of the material discussed in class, and also, Alternatively, we can п¬Ѓnd Q Вј P*1 by forming the matrix M ВјВЅP;I& and row reducing M to row canonical form: M Вј 1211 00 0120 10 1220 01 2 6 4 3 7 5+ 10 0*2 *23.

Math 217 Summary of Change of Basis and All That

Linear Algebra/Changing Representations of Vectors. Given that the change of basis has once the basis matrix and once its inverse, these objects are said to be 1-co, 1-contra-variant. The matrix of an endomorphism [ edit ] An important case of the matrix of a linear transformation is that of an endomorphism , that is, a linear map from a vector space V to itself: that is, the case that W = V ., Proof (1 of 2) I Let V is a п¬Ѓnite-dimensional inner product space, let Band B0be two orthonormal bases for V, and let P be the transition matrix from B0to B..

Given that the change of basis has once the basis matrix and once its inverse, these objects are said to be 1-co, 1-contra-variant. The matrix of an endomorphism [ edit ] An important case of the matrix of a linear transformation is that of an endomorphism , that is, a linear map from a vector space V to itself: that is, the case that W = V . EECS 16A Designing Information Devices and Systems I Spring 2016 O cial Lecture Notes Note 21 Introduction In this lecture note, we will introduce the last topics of this semester, change of basis …

Change of Coordinates with Orthonormal Bases (pages 325-329) Because it is so easy to nd the coordinates with respect to an orthonormal basis B= f~v 1;:::;~v ng, we will easily be able to nd the change of coordinates matrix from the standard basis Sto B. In fact, it will be easier than we could have hoped! After all, the change of coordinates matrix Q from Sto Bsatis es the following: Q = [~e The power of linear algebra in practice stems from the fact that we can choose bases so as to simplify the form of the matrix representing the object in question. We will see several such “canonical form theorems” in the notes.

To gain a good undertanding of a self linear transformation, we find a basis (a suitable invertible matrix P) so that the the transformation behaves particularly nice with respect to the basis (the similarity change via P gives a diagonal matrix). Assignment 4/MATH 247/Winter 2010 . Due: Tuesday, February 9 . First, you find, in Part 1, a summary of some of the material discussed in class, and also

Final Exam Linear Algebra, Dave Bayer, December 16, 1999 Please work only one problem per page, starting with the pages provided, and number all continuations clearly.Only work which can be found in this way will be graded. Outline: 14. EXAMPLES OF CHANGE OF BASIS AND MATRIX TRANSFORMATIONS. QUADRATIC FORMS. 14.1 Examples of change of basis 14.1.1 Representation of a 2D vector in a …

Given that the change of basis has once the basis matrix and once its inverse, these objects are said to be 1-co, 1-contra-variant. The matrix of an endomorphism [ edit ] An important case of the matrix of a linear transformation is that of an endomorphism , that is, a linear map from a vector space V to itself: that is, the case that W = V . Given linear mapping and bases, determine the transformation matrix and the change of basis Hot Network Questions Are elves in Middle Earth mortal or immortal?

Change of basis • Let B denote a matrix whose columns form an orthonormal basis for a vector space W If B is full rank (n x n), then we can get back to the original basis through multiplication by B. Change of basis • Let B denote a matrix whose columns form an orthonormal basis for a vector space W Vector of projections of v along each basis vector . Orthogonal matrix • In this case Given coordinates of a point in one basis, we will be able to obtain the coordinates of that point in another basis by applying the change of coordinates matrix to it. § A3 .2 Change of Basis Basics Let us first recall a few basic facts about bases and change of basis matrices.

EECS 16A Designing Information Devices and Systems I Spring 2016 O cial Lecture Notes Note 21 Introduction In this lecture note, we will introduce the last topics of this semester, change of basis … 27/12/2015 · The basis are denoted A and B. Similar to the previous video, we again compute a "change-of-coordinates" matrix that can transform a vector written in each basis …

To gain a good undertanding of a self linear transformation, we find a basis (a suitable invertible matrix P) so that the the transformation behaves particularly nice with respect to the basis (the similarity change via P gives a diagonal matrix). Change of basis. Linear functions and their matrices 5.1 Linear maps from Rk to Rn See [D]Section 4.2. The important thing is thatany linear map f: Rk! Rn can be expressed by a matrix multiplication. The matrix associated with f is Mf = f(e1) f(e2)::: f(ek)! i.e. the vectors f(ei) form the columns of Mf,wheree1;e2;:::is the standard basis in Rk 5.2 Vectorspaces So far we always played with

Change of basis. Linear functions and their matrices 5.1 Linear maps from Rk to Rn See [D]Section 4.2. The important thing is thatany linear map f: Rk! Rn can be expressed by a matrix multiplication. The matrix associated with f is Mf = f(e1) f(e2)::: f(ek)! i.e. the vectors f(ei) form the columns of Mf,wheree1;e2;:::is the standard basis in Rk 5.2 Vectorspaces So far we always played with A) Find the change of basis matrix for converting from the standard basis to the basis B. I have never done anything like this and the only examples I can find online basically tell me how to do the change of basis for "change-of-coordinates matrix from B to C".

Change of basis for matrices for linear transformations. The ma-trix Athat represents a linear transformation L: V в†’ V relative to a basis E= [v Alternatively, we can п¬Ѓnd Q Вј P*1 by forming the matrix M ВјВЅP;I& and row reducing M to row canonical form: M Вј 1211 00 0120 10 1220 01 2 6 4 3 7 5+ 10 0*2 *23

Change of Basis Let $V$ be a vector space and let $S = \{{\bf v_1,v_2, \ldots, v_n}\}$ be a set of vectors in $V$. Recall that $S$ forms a basis for $V$ if the basis and writing it in terms of the second basis, we would multiply by the inverse of the basis change matrix: 1 Note this istrue forcomposition oflineartransformations forany sizevectorspaces,

Basis & Change of Basis Christopher Eur November 5, 2013 In this article we develop notion of basis and change of basis matrix in a more traditional way is the change of basis matrix of the basis. In order to understand this relationship In order to understand this relationship better, it is convenient to take it as a de nition and then study it abstractly.

Worksheet 19: Change of basis Assume that V is some vector space and dimV = n<1. Let B= f~b 1;:::;~b ngand C= f~c 1;:::;~c ngbe two bases of V. For any vector ~v2V, let [~v] Band [~v] Cbe its coordinate vectors with respect to the bases Band C, respectively. These vectors are related by the formula [~v] C= P C B[~v] B: (1) Here P C Bis the change of coordinates matrix from Bto C, given … Change of coordinates Math 130 Linear Algebra D Joyce, Fall 2015 The coordinates of a vector v in a vector space V with respect to a basis = fb 1;b

Caution: Do not confuse the roles of the bases Aand Bin Definition II. Of course, we could also convert from A-coordinates to B-coordinates in the same way, but the matrix that does this, denoted S Theorem. A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. i.e. P 1 = PT: Example Consider R3 with the orthonormal basis S= 8 >> < >>: u 1 = 0 B B @ p2 6 p 1 6 p 1 6 1 C C A;u 2 = 0 B B @ 0 p 2 p 2 1 C C A;u 3 = 0 B B @ 1 3 p 3 p 3 1 C C A 9 >> = >>;: Let Rbe the standard basis fe 1;e 2;e 3g. Since we are changing from the standard basis to a new basis, then

This means that any square, invertible matrix can be seen as a change of basis matrix from the basis spelled out in its columns to the standard basis. This is a natural consequence of how multiplying a matrix by a vector works by linearly combining the matrix's columns . n form a basis set in a vector space if, and only if, each vector in the space can be represented uniquely as a sum of scalar multiples of the basis vectors, as in (2). There are two requirements

We finish this subsection by recognizing that the change of basis matrices are familiar. Lemma 1.4 A matrix changes bases if and only if it is nonsingular. Proof For one direction, if left-multiplication by a matrix changes bases then the matrix represents an invertible function, simply because the Change of basis in polynomial interpolation W. Gander Institute of Computational Science ETH Zurich CH-8092 Zurich Switzerland SUMMARY Several representations for the interpolating polynomial exist: Lagrange, Newton, orthogonal polynomials etc. Each representation is characterized by some basis functions. In this paper we investigate the transformations between the basis functions which …

Change of Basis University of Connecticut

change of basis matrix pdf

Example of Change of Basis YouTube. Change of Coordinates with Orthonormal Bases (pages 325-329) Because it is so easy to nd the coordinates with respect to an orthonormal basis B= f~v 1;:::;~v ng, we will easily be able to nd the change of coordinates matrix from the standard basis Sto B. In fact, it will be easier than we could have hoped! After all, the change of coordinates matrix Q from Sto Bsatis es the following: Q = [~e, Theorem. A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. i.e. P 1 = PT: Example Consider R3 with the orthonormal basis S= 8 >> < >>: u 1 = 0 B B @ p2 6 p 1 6 p 1 6 1 C C A;u 2 = 0 B B @ 0 p 2 p 2 1 C C A;u 3 = 0 B B @ 1 3 p 3 p 3 1 C C A 9 >> = >>;: Let Rbe the standard basis fe 1;e 2;e 3g. Since we are changing from the standard basis to a new basis, then.

change of basis – Problems in Mathematics yutsumura.com

change of basis matrix pdf

Lecture 2c Change of Coordinates with Orthonormal Bases. Final Exam Linear Algebra, Dave Bayer, December 16, 1999 Please work only one problem per page, starting with the pages provided, and number all continuations clearly.Only work which can be found in this way will be graded. 11/09/2016 · 14 videos Play all Essence of linear algebra 3Blue1Brown 5 Levels S1 • E6 Quantum Computing Expert Explains One Concept in 5 Levels of Difficulty WIRED - ….

change of basis matrix pdf


Theorem. A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. i.e. P 1 = PT: Example Consider R3 with the orthonormal basis S= 8 >> < >>: u 1 = 0 B B @ p2 6 p 1 6 p 1 6 1 C C A;u 2 = 0 B B @ 0 p 2 p 2 1 C C A;u 3 = 0 B B @ 1 3 p 3 p 3 1 C C A 9 >> = >>;: Let Rbe the standard basis fe 1;e 2;e 3g. Since we are changing from the standard basis to a new basis, then n form a basis set in a vector space if, and only if, each vector in the space can be represented uniquely as a sum of scalar multiples of the basis vectors, as in (2). There are two requirements

of basis vectors and are related by linear transformations to the components in another basis. When such transformations conserve the lengths of vectors, they are said to be EECS 16A Designing Information Devices and Systems I Spring 2016 O cial Lecture Notes Note 21 Introduction In this lecture note, we will introduce the last topics of this semester, change of basis …

Proof (1 of 2) I Let V is a finite-dimensional inner product space, let Band B0be two orthonormal bases for V, and let P be the transition matrix from B0to B. The power of linear algebra in practice stems from the fact that we can choose bases so as to simplify the form of the matrix representing the object in question. We will see several such “canonical form theorems” in the notes.

11/09/2016 · 14 videos Play all Essence of linear algebra 3Blue1Brown 5 Levels S1 • E6 Quantum Computing Expert Explains One Concept in 5 Levels of Difficulty WIRED - … 27/12/2015 · The basis are denoted A and B. Similar to the previous video, we again compute a "change-of-coordinates" matrix that can transform a vector written in each basis …

Proof (1 of 2) I Let V is a п¬Ѓnite-dimensional inner product space, let Band B0be two orthonormal bases for V, and let P be the transition matrix from B0to B. This means that any square, invertible matrix can be seen as a change of basis matrix from the basis spelled out in its columns to the standard basis. This is a natural consequence of how multiplying a matrix by a vector works by linearly combining the matrix's columns .

27/12/2015 · The basis are denoted A and B. Similar to the previous video, we again compute a "change-of-coordinates" matrix that can transform a vector written in each basis … Change of Basis 2: Matrices In this worksheet we will calculate what matrices look like in various bases. Whereas we need only one basis to write down a vector, we need two bases to write down a matrix. Loosely speaking we need a basis for the rows and a basis for the columns. We will explain this more exactly in a minute. For our calculations we will use the space of 3-D vectors R3, with the

Change of basis for matrices for linear transformations. The ma-trix Athat represents a linear transformation L: V в†’ V relative to a basis E= [v Alternatively, we can п¬Ѓnd Q Вј P*1 by forming the matrix M ВјВЅP;I& and row reducing M to row canonical form: M Вј 1211 00 0120 10 1220 01 2 6 4 3 7 5+ 10 0*2 *23

Math 28S Coordinate changes Fall 2011 1 Change of basis matrices Suppose V is a vector space over eld F with dimension n. Then, we know that V ˘=Fn Chapter & Page: 5–2 Change of Basis and AA† = 3 5 4 5 i 4 5 i 3 5 3 5 −4 5 i −4 5 i 3 5 = ··· = I . So A is unitary. Obviously, for a matrix to be unitary, it must be square.

Outline: 14. EXAMPLES OF CHANGE OF BASIS AND MATRIX TRANSFORMATIONS. QUADRATIC FORMS. 14.1 Examples of change of basis 14.1.1 Representation of a 2D vector in a … Change of basis for matrices for linear transformations. The ma-trix Athat represents a linear transformation L: V → V relative to a basis E= [v

basis and writing it in terms of the second basis, we would multiply by the inverse of the basis change matrix: 1 Note this istrue forcomposition oflineartransformations forany sizevectorspaces, Conversely, if V is any vector space and if v1,...,v n is a basis for V, then we define the matrix [H] v 1,...,v n for H with respect to this basis to be the matrix

n form a basis set in a vector space if, and only if, each vector in the space can be represented uniquely as a sum of scalar multiples of the basis vectors, as in (2). There are two requirements The goal of these notes is to provide an apparatus for dealing with change of basis in vector spaces, matrices of linear transformations, and how the matrix changes when the basis changes.

27/12/2015 · The basis are denoted A and B. Similar to the previous video, we again compute a "change-of-coordinates" matrix that can transform a vector written in each basis … Math 20F Linear Algebra Lecture 16 2 Slide 3 ’ & $ % Basis and components De nition 2 (Dimension) A vector space V has dimension nif the maximum number of l.i. vectors is n.

Change of Basis Let $V$ be a vector space and let $S = \{{\bf v_1,v_2, \ldots, v_n}\}$ be a set of vectors in $V$. Recall that $S$ forms a basis for $V$ if the Complete Change of basis matrix - Mathematics, Engineering chapter (including extra questions, long questions, short questions) can be found on EduRev, you can check out Engineering Mathematics lecture & lessons summary in the same course for Engineering Mathematics Syllabus. EduRev is like a wikipedia just for education and the Change of basis matrix - Mathematics, Engineering images and

Change of basis in polynomial interpolation W. Gander Institute of Computational Science ETH Zurich CH-8092 Zurich Switzerland SUMMARY Several representations for the interpolating polynomial exist: Lagrange, Newton, orthogonal polynomials etc. Each representation is characterized by some basis functions. In this paper we investigate the transformations between the basis functions which … Definition. A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C) is a linearly independent subset of V that spans V.