Basic Notions of Linear Algebra

Linear algebra deals with finite dimensional real or complex spaces, called R or Cn for any positive integer n. A typical n-vector i e K or Cn has the form of a row 

With xT denoting the transposed vector of x, i.e., the row vector for x if x is a column vector and vice versa, the dot product can be visualized in the form of a matrix product as a row times a column vector 

This concept allows us to express every linear function / R — Rm as a constant matrix times vector product. Here a matrix A G Rm,n is an m by n rectangular array of numbers ay in R or C where m counts the number of rows in A, while n is the number of A s columns. 

An m by n matrix A can be expressed three ways in terms of its entries ay, where the first index i denotes the row that ay appears in and the second index j denotes the column or a matrix A = A hn can be denoted by its n columns cy. c RTO or by its m rows ry. rTO Rn as depicted above. 

We define the standard unit vectors et of Rn to be the n-vector of all zero entries, except for position i which is one, such as e = (1, 0, 0, 0)T E R4 or en i = (0, 0,. .., 1, 0)T Rn in column notation. With this notation every linear function / Rn — Rm can be represented as a constant matrix times vector product f(x) = Ax for 

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