By Alan Macdonald

This textbook for the 1st undergraduate linear algebra path offers a unified remedy of linear algebra and geometric algebra, whereas overlaying lots of the traditional linear algebra topics.Geometric algebra is an extension of linear algebra. It complements the remedy of many linear algebra subject matters. And geometric algebra does a lot more.Geometric algebra and its extension to geometric calculus unify, simplify, and generalize mammoth components of arithmetic that contain geometric rules. they supply a unified mathematical language for plenty of parts of physics, laptop technology, and different fields.The booklet can be utilized for self learn by way of these ok with the theorem/proof type of a arithmetic text.

This is a fourth corrected printing. bankruptcy 6 has been rearranged. there's a new bankruptcy at the conformal version. The bankruptcy is additionally to be had for obtain on the book's net page.

Visit the book's online page at faculty.luther.edu/~macdonal/laga to obtain its desk of contents, preface, and index.

There is a sequel, *Vector and Geometric Calculus, * by way of a similar author.

See faculty.luther.edu/~macdonal/vagc/

I commend Alan Macdonald for his first-class booklet! His exposition is fresh and spare. He has performed a very good activity of engineering a steady transition from normal perspectives of linear algebra to the viewpoint of geometric algebra. The e-book is satisfactorily traditional to be followed as a textbook through an adventurous instructor with no getting flack from colleagues. but it results in gemstones of geometric algebra which are prone to pride considerate scholars and shock even the main skilled instructors.

-- David Hestenes, individual examine Professor, Arizona nation college

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**Extra resources for Linear and Geometric Algebra**

**Example text**

Example: A — 1 -2 2 1 -1 1 This is a 2 x 3 matrix: it has 2 rows and 3 columns. ) Denote th e entry in th e i th row and j th column by a ^ . Thus <221 = —2. 1) A \^ij] a 21 <222 a 23 Each row of th e matrix is a vector in R 3, a row vector. E ach column is a vector in M2 displayed vertically, a column vector. Some te x ts use different notations for a vector displayed as a row and as a column. We will not. Note: T he plural of “m atrix ” is “m atrices” ( not “m atrix es” ). C 34 hapter 3: M a tr ic es A r ith m e tic O perations We introduce the three basic arithm etic operations on matrices: scalar mul tiplication, m atrix addition, and m atrix multiplication.

Suppose that the product A B of matrices A and B exists. Show that IC(B)CIC(AB). b. Show that IC(A) = 1C(A*A). Hint: To show that IC(A*A) C IC(A), start x e KL(A*A) = » ■■■. c. Show that if fC(A) = {0}, then the square matrix A*A has an inverse. 7. a. Solve in the form Eq. 17): w —2 x + 3y + 4z = 2 2w + 7y —5z = 9 w + x + y + z = 8. Use the p r i n t r r e f method of SymPy. b. Remove the third equation in Part (a) and solve. 8. Solve 2x — 2y + z + 3w = 3, 2x + y — 2z — 4, x + 2y — z —2w = 1 in the form Eq.

23 (Centralizer). Let A be an n x n matrix. The centralizer C(A) of A is the set of all ii x n matrices which commute with A. a. Prove that C(A) is a subspace of the vector space o f n x n matrices. b. Prove that C(A) is closed under matrix multiplication. 24 (Change of basis). This problem shows how the representation [v]g changes when B changes. Let B = {bj} and B' = {b'} be bases for an 71dimensional vector space. Then [b^g- denotes the coordinates of b* 6 B with respect to the B’ basis. Place these coordinates as columns of a matrix [i]g'g: [ijs'B = [ [t»i]s' [b2]g< ••• [bn]B/].