Chapter 4 Vector Spaces Linear Algebra

Ch04_2 Definition 1. Let be a sequence of n real numbers. The set of all such sequences is called n-space (or n-dimensional. space) and is denoted R n. u 1 is the first component of. u 2 is the second component and so on. Example 1 R 2 is the collection of all sets of two ordered real numbers. For example, (0, 0), (1, 2) and (-2, -3) are elements of R 2. R 3 is the collection of all sets of three ordered real numbers. For example, (0,0, 0) and (-1,3, 4) are elements of R The vector Space R n

Ch04_3 Definition 2. Let be two elements of R n. We say that u and v are equal if u 1 = v 1, …, u n = v n. Thus two elements of R n are equal if their corresponding components are equal. Definition 3. Let be elements of R n and let c be a scalar. Addition and scalar multiplication are performed as follows: Addition: Scalar multiplication :

Ch04_4 ► The set R n with operations of componentwise addition and scalar multiplication is an example of a vector space, and its elements are called vectors. We shall henceforth interpret R n to be a vector space. ( We say that R n is closed under addition and scalar multiplication). ► In general, if u and v are vectors in the same vector space, then u + v is the diagonal of the parallelogram defined by u and v. Figure 4.1

Ch04_5 Example 2 Let u = ( –1, 4, 3) and v = ( –2, –3, 1) be elements of R 3. Find u + v and 3u. Solution: Example 3 Figure 4.2 In R 2, consider the two elements (4, 1) and (2, 3). Find their sum and give a geometrical interpretation of this sum. we get (4, 1) + (2, 3) = (6, 4). The vector (6, 4), the sum, is the diagonal of the parallelogram.

Ch04_6 Example 4 Figure 4.3 Consider the scalar multiple of the vector (3, 2) by 2, we get 2(3, 2) = (6, 4) Observe in Figure 4.3 that (6, 4) is a vector in the same direction as (3, 2), and 2 times it in length.

Ch04_7 Zero Vector The vector (0, 0, …, 0), having n zero components, is called the zero vector of R n and is denoted 0. Negative Vector The vector (–1)u is writing –u and is called the negative of u. It is a vector having the same length (or magnitude) as u, but lies in the opposite direction to u. Subtraction Subtraction is performed on element of R n by subtracting corresponding components. u uu

Ch04_8 Theorem 4.1 Let u, v, and w be vectors in R n and let c and d be scalars. (a)u + v = v + u (b)u + (v + w) = (u + v) + w (c)u + 0 = 0 + u = u (d)u + (–u) = 0 (e) c(u + v) = cu + cv (f)(c + d)u = cu + du (g) c(du) = (cd)u (h) 1u = u Figure 4.4 Commutativity of vector addition u + v = v + u

Ch04_9 Example 5 Let u = (2, 5, –3), v = ( –4, 1, 9), w = (4, 0, 2) in the vector space R 3. Determine the vector 2u – 3v + w. Solution

Ch04_10 Column Vectors additionscalar multiplication We defined addition and scalar multiplication of column vectors in R n in a componentwise manner: and Row vector: Column vector:

Ch04_11 Homework Exercise set 1.3 page 32: 3, 5, 7, 9.

Ch04_ Dot Product, Norm, Angle, and Distance Definition Let be two vectors in R n. The dot product of u and v is denoted u.v and is defined by. The dot product assigns a real number to each pair of vectors. Example 1 Find the dot product of u = (1, –2, 4) and v = (3, 0, 2) Solution

Ch04_13 Properties of the Dot Product Let u, v, and w be vectors in R n and let c be a scalar. Then 1.u.v = v.u 2.(u + v).w = u.w + v.w 3. cu.v = c(u.v) = u.cv 4.u.u  0, and u.u = 0 if and only if u = 0

Ch04_14 Norm of a Vector in R n Definition The norm (length or magnitude) of a vector u = (u 1, …, u n ) in R n is denoted ||u|| and defined by Note: The norm of a vector can also be written in terms of the dot product Figure 4.5 length of u

Ch04_15 Definition A unit vector is a vector whose norm is 1. If v is a nonzero vector, then the vector is a unit vector in the direction of v. This procedure of constructing a unit vector in the same direction as a given vector is called normalizing the vector. Find the norm of each of the vectors u = (1, 3, 5) of R 3 and v = (3, 0, 1, 4) of R 4. Solution Example 2

Ch04_16 Example 3 Solution (a)Show that the vector (1, 0) is a unit vector. (b)Find the norm of the vector (2, –1, 3). Normalize this vector.

Ch04_17 Angle between Vectors ( in R 2 ) ← Figure 4.6 The law of cosines gives:

Ch04_18 Definition Let u and v be two nonzero vectors in R n. The cosine of the angle  between these vectors is Example 4 Determine the angle between the vectors u = (1, 0, 0) and v = (1, 0, 1) in R 3. Solution Angle between Vectors (in R n )

Ch04_19 Definition Two nonzero vectors are orthogonal if the angle between them is a right angle. Two nonzero vectors u and v are orthogonal if and only if u.v = 0. Theorem 4.2 Proof Orthogonal Vectors

Ch04_20 Solution Example 5 Show that the following pairs of vectors are orthogonal. (a)(1, 0) and (0, 1). (b)(2, –3, 1) and (1, 2, 4).

Ch04_21 Note (1, 0), (0,1) are orthogonal unit vectors in R 2. (1, 0, 0), (0, 1, 0), (0, 0, 1) are orthogonal unit vectors in R 3. (1, 0, …, 0), (0, 1, 0, …, 0), …, (0, …, 0, 1) are orthogonal unit vectors in R n.

Ch04_22 Example 6 Determine a vector in R 2 that is orthogonal to (3, –1). Show that there are many such vectors and that they all lie on a line. Solution Figure 4.7

Ch04_23 Theorem 4.3 Let u and v be vectors in R n. (a)Triangle Inequality: ||u + v||  ||u|| + ||v||. (a)Pythagorean theorem : If u.v = 0 then ||u + v|| 2 = ||u|| 2 + ||v|| 2. Figure 4.8(a)Figure 4.8(b)

Ch04_24 Distance between Points Let be two points in R n. The distance between x and y is denoted d(x, y) and is defined by Note: We can also write this distance as follows. Example 7. Determine the distance between the points x = (1, – 2, 3, 0) and y = (4, 0, – 3, 5) in R 4. Solution x y xyxy

Ch04_25 Homework Exercise set 1.5 pages 47 to 48: 3, 7, 8, 9, 11, 13, 16, 17, 26. Exercise 36 Let u and v be vectors in R n. Prove that ||u|| = ||v|| if and only if u + v and u  v are orthogonal.