is in plane V. However, it may be more convenient to introduce a plane coordinate system in V.

Transcription

1 .4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5 is in plane V. However, it may be more convenient to introduce a plane coordinate system in V. Consider any two vectors in plane V that aren t parallel, e.g. v and v 2 2

2 See Figure, where we label the new axes c and c 2, with the new coordinate grid defined by vectors v and v 2. Note that the c - c 2 coordinates of vector v c is and the coordinates of vector c v 2 is, respectively. For a vector x in plane V, we can find the scalars c and c 2 such that For example, x x c v + c 2 v

3 Therefore, the c c 2 coordinates of x are c c 2 2 See Figure. Let s denote the basis v, v 2 of V by (Fraktur ). Then, the coordinate vector of x with respect to is denoted by x : If x 5, then x 2

4 Definition.4. Coordinates in a subspace of R n Consider a basis of a subspace V of R n, consisting of vectors v, v 2,..., v m. Any vector x in V can be written uniquely as xc v +c 2 v c m v m The scalars c, c,..., c m are called the - coordinates of x, and the vector c c 2... c m is called the -coordinate vector of x, denoted by x. Note that where S x S x v v 2... v m, an n m matrix. 4

5 EXAMPLE 2 Consider the basis of R 2 consisting of vectors v and v 2 0 a. If x, find x 0 b. If x 2, find x Solution a. To find the coordinates of vector x, we need to write x as a linear combination of the basis vectors: x c v + c 2 v 2, or 0 0 c + c 2 Alternatively, we can solve the equation x S x for x c c2 x 5

6 x 0 S x b. y definition of coordinates, x means that x 2 v +() v 2 2 +() 2 7 Alternatively, use the formula x S x 2 7 6

7 EXAMPLE Let L be the line in R 2 spanned by vector Let T be the linear transformation from R 2 to R 2 that projects any vector orthogonally onto line L, as shown in Figure 5... In x x 2 coordinate system (See Figure 5): Sec 2.2 (pp. 59). 2. In c c 2 coordinate system (See Figure 6): T transforms vector c c 2 into c 0 That is, T is given by the matrix 0 0 c c, since c 2. The transforms from x into T ( x) is called the -matrix of T : T ( x) x 7

8 Definition.4.2 The -matrix of a linear transformation Consider a linear transformation T from R n to R n and a basis of R n. The n n matrix that transforms x into T ( x) is called the -matrix of T : for all x in R n. T ( x) x Fact.4. The columns of the -matrix of a linear transformation Consider a linear transformation T from R n to R n and a basis of R n consisting of vectors v, v 2,..., v n. Then, the -matrix of T is T ( x ) T ( x2 )... T ( x n ) That is, the columns of are the -coordinate vectors of T( v ), T( v 2 ),..., T( v n ). 8

9 EXAMPLE 4 Consider two perpendicular unit vectors v and v 2 in R. Form the basis v, v 2, v v v 2 of R ; let s denote this basis by. Find the - matrix of the linear transformation T( x) v x. (see Exercise 2.: 44 on pp. 49, a b a 2 b a b 2 a 2 b 2 a b a b ) a b a b 2 a 2 b Solution Use Fact.4. to construct column by column: T ( x ) T ( x2 )... T ( x n ) v v v v 2 v v 0 v v

10 EXAMPLE 5 Let T be the linear transformation from R 2 to R 2 that projects any vector orthogonally onto the line L spanned by. In Example, we found that the matrix of T with respect to the basis consisting of and is What is the relation ship between and the standard matrix A of T (such that T( x)a x)? Solution Recall from Definition.4. that x S x, where S and consider the following diagram: (Figure 7) 0

11 Note that T( x)as x and also T( x)s x, so that AS x S x for all x. Thus, ASS and ASS Now we can find the standard matrix A of T : ASS ( 0 ) Alternatively, we could use Fact to construct matrix A. The point here was to explore the relationship between matrices A and.

12 Fact.4.4 Standard matrix versus -matrix of a linear transformation Consider a linear transformation T from R n to R n and a basis of R n consisting of vectors v, v 2,..., v n. Let be the -matrix of T and let A be the standard matrix of T (such that T( x)a x). Then, AS S, S AS, and A SS, where S v v 2... v m Definition.4.5 Similar matrices Consider two n n matrices A and. We say that A is similar to if there is an invertible matrix S such that ASS, or S AS

13 EXAMPLE 6 2 Is matrix A 4 similar to 5 0 0? Solution We are looking for a matrix S that ASS, or x + 2z 4x + z y + 2t 4y + t These equations simplify to x y z t 5x y 5z t. such z 2x, t y, so that any invertible matrix of the form S x 2x y y does the job. Note that det(s)-xy. Matrix S is invertible if det(s) 0 (i.e.,if neither x nor y is zero). 2

14 EXAMPLE 7 Show that if matrix A is similar to, then its power A t is similar to t for all positive integers t. (That is, A 2 is similar to 2, A is similar to, etc.) Solution We know that S AS for some invertible matrix S. Now, t (S AS)(S AS)...(S AS)(S AS) } {{ } t times S A t S, proving our claims. Note the cancellation of many terms of the form SS.

15 Fact.4.6 Similarity is an equivalence relation. An n n matrix A is similar to itself (Reflexivity). 2. If A is similar to, then is similar to A (Symmetry).. If A is similar to and is similar to C, then A is similar to C (Transitivity). Proof A is similar to : P AP is similar to C: C Q Q, then C Q Q Q P AP Q (P Q) A(P Q) that is, A is similar to C by matrix P Q. Homework Exercise.4: 5, 6, 9, 0,, 4, 9,, 9 4