Span and Linear Independence

Transcription

1 Span and Linear Independence It is common to confuse span and linear independence, because although they are different concepts, they are related. To see their relationship, let s revisit the previous example (from the last set of lecture notes) which involved the set of two vectors {v, v 2 }. This set is linearly dependent since v 2 is a multiple of v that is, v 2 is in Span{v }. This connection between span and linear independence generalizes to sets of more than two vectors. Theorem: Let {v, v 2,..., v p } be a set of at least two vectors in R n. Then the set is linearly dependent if and only if at least one of the vectors in the set is in the span of the other vectors. Example: Let v =, v 2 =, v 3 = 2 3. Since v 3 = 2v + v 2, v 3 is in Span{v, v 2 }. Thus, {v, v 2, v 3 } is linearly dependent. Warning: Be careful not to be misled by the preceding theorem. It does not imply that every vector in a linearly dependent set is a linear combination of the other vectors. In other words, if {v, v 2, v 3 } is a linearly dependent set where v is in Span{v 2, v 3 }, this does not mean that v 2 is in Span{v, v 3 } and v 3 is in Span{v, v 2 }. Theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v, v 2,..., v p } in R n is linearly dependent if p > n. Example: Let v = 2, v 2 = 3 2, v 3 = 5 2, v 4 = 2 6 Since the number of vectors is greater than the number of components in each vector, the set {v, v 2,..., v p } is linearly dependent by the preceding theorem. This means that at least one of the vectors is a linear combination of the other vectors. In particular, v 3 = 2v +v 2. However, v 4 is not a linear combination of v, v 2, and v 3 (verify this!) Warning: The preceding theorem tells us nothing about the case when n p. In this situation, the set could either be linearly dependent or linearly independent.

2 Theorem: If a set S = {v, v 2,, v p } in R n contains the zero vector, then the set is linearly dependent. Why is this theorem true? Let s prove this. Proof: Suppose that one of the vectors is the zero vector, say v p =. We need to show that there exist scalars x, x 2,..., x p not all zero such that x v + x 2 v x p v p = We can let x = x 2 = = x p = and let x p be any nonzero scalar. Doing so establishes the linear dependence of S. Example: Find h so that the vectors are linearly dependent. v = 2 2, v 2 = 2, v 3 = 4 3 h Solution: Let s set up the augmented matrix v v 2 v 3 ] and reduce it to echelon form: h h h Recall that if every column of the coefficient matrix has a pivot, then the set of vectors {v, v 2, v 3 } is linearly independent. Note that the first two columns of the echelon matrix above are pivot columns. Since we don t want the last column to be a pivot column, we would need to force the entry 3 + h to be zero. Thus, when h = 3, {v, v 2, v 3 } is linearly dependent.

3 Sections.8 &.9: Linear Transformations In your past algebra courses, you studied the concept of a function. Recall that a function, f, is a rule that assigns to each input (variable x) a unique output (variable y = f(x)). For example, let f(x) = 2x. If the input is x =, then the output is f() = 2. This is an elementary idea, but let s stop and think about what the function actually did to the input. The function took the input and doubled it to produce an output of 2. In fact, if we plug in any input, the function will always double it since that is exactly what the rule is. The general idea here is that each number (input) that you enter into a function is transformed into a different number (output) according to what the function (the rule) is. Now how does this idea extend to the world of matrices and vectors? Let s look at an example. Example: Let s define Ax = b where x and b are vectors in R 2 and 2 A =. We know that for each x in R 2 that we plug in, we obtain a new vector b in R 2. Therefore, we can think of T (x) = Ax as being the function, and x and b = T (x) as the input and output, respectively. Let s plug in x = and see what we get. It follows that 3] 2 3] }{{}}{{} A x Notice the effect that the matrix A has on the vector x geometrically. The matrix stretches 5 x = and rotates it to produce a new vector b =. (See Figure.) In general, all that 3] 2] a matrix A does to a vector x (when A is multiplied to x) is that A stretches (or compresses) x and then relocates it (by rotation or reflection). = 5 2] }{{} b Figure : Geometric description of matrix-vector multiplication (Ax = b).

4 The matrix A in the previous example is an example of a transformation. Definition: A transformation (or function or mapping) T from R n to R m is a rule that assigns to each vector x in R n a vector T (x) in R m. The set R n is called the domain of T, and R m is called the codomain of T. Notation: The notation T : R n R m simply means that T is a function that transforms vectors in R n (domain) to vectors in R m (codomain). Definition: For x in R n, the vector T (x) in R m is called the image of x under T. The set of all images T (x) is called the range of T. Matrix Transformations When T is defined to be T (x) = Ax, where A is an m n matrix and x is in R n, then T is called a matrix transformation. The transformation itself can be denoted by x Ax, which simply means that x is being mapped to the vector Ax. The matrix A is called the standard matrix for the linear transformation T. Note: If the size of A is m n and the size of x is n, then the size of b = Ax is m. Example: Define T : R 3 R 2 to be a transformation that projects every vector in R 3 onto the xy-plane. What is the image of x = 2 under T? 3 Solution: Since we are projecting x onto the xy-plane, we are expecting that the third component (the z-coordinate) of the image is zero. Note that the codomain is R 2, which means that we will need to express the output as a two-component vector. Thus, T 2 = 2] 3 The standard matrix that does this particular transformation is A =

5 Let s verify this. Ax = 2 = 3 () + (2) + (3) = () + (2) + (3) 2] Example: Let A = ] and u = (a) Define T : R 3 R 2 by T (x) = Ax. Determine T (u). Solution: Applying the same procedure as in the previous example, we obtain 2 T (u) = Au =. 7] 3 (b) Find a vector x whose image under T is. Is x unique? 4 3 Solution: Let b =. We wish to find a vector x such that T (x) = Ax = b. Note 4 that Ax = x x 2 = x = x = 3, x 2 = 4, x 3 = free Therefore, every vector x in R 3 that maps to b has the form x = x 3 }{{} p One such vector is p. Since the system has a free variable, x is not unique. Note that we could have simply used the fact that A projects every vector in R 3 onto the xy-plane. Thus, any vector x in R 3 with an x-coordinate of 3 and a y-coordinate of 4 works. (c) Determine if b = 3 is in the range of T. 4 Solution: We showed in part (b) that there exists an x in R 3 such that T (x) = b. Thus, b is in the range of T.

6 Example: Let A = (a) Find all x in R 4 that are mapped into the zero vector by the transformation x Ax. Solution: Note that the augmented matrix A ] has the following reduced echelon form 4 R = 3 Therefore, the system of equations is x 4x 3 = x 2 3x 3 = x 4 = where x 3 is a free variable. It follows that every x in R 4 that is mapped to has the form x 4x 3 4 x = x 2 x 3 = 3x 3 x 3 = x 3 3. x 4 (b) Determine if the vector b = is in the range of x Ax. Solution: To show that b is in the range of the transformation, it suffices to show that the columns of A span R 3. This is equivalent to showing that every row of A has a pivot. The coefficient matrix R in part (a) has a pivot in every row (and therefore this must also be the case for A). Hence, the columns of A span R 3, and so b must be in the range of the transformation.

7 Linear Transformations Definition: A transformation (or mapping) T is linear if: (i) T (u + v) = T (u) + T (v) for all u, v in the domain of T ; (ii) T (cu) = ct (u) for all scalars c and all u in the domain of T. Fact: Every matrix transformation is a linear transformation. Fact: If T is a linear transformation, then T () = and T (cu + dv) = ct (u) + dt (v) for all vectors u, v in the domain of T and all scalars c, d. In general, T (c v + + c p v p ) = c T (v ) + + c p T (v p ) which is known as a superposition principle. Dilation and Contraction Operators If k is a nonnegative scalar, then the transformation defined by T (x) = kx on R 2 or R 3 is called a contraction with factor k if k and a dilation with factor k if k >. The geometric effect of a contraction is to compress each vector by a factor of k, and the effect of a dilation is to stretch each vector by a factor of k (See Figure 2). A contraction compresses R 2 or R 3 uniformly toward the origin from all directions, and a dilation stretches R 2 or R 3 uniformly away from the origin in all directions. Figure 2: A contraction when k (left) and a dilation when k > (right).

8 Rotation Operators An operator that rotates each vector in R 2 through a fixed angle θ is called a rotation operator on R 2. Let s look at an example. Example: Define T : R 2 R 2 by x T (x) = x 2 }{{}}{{} A x Let s play around with this transformation. If we plug in x = ( ) T = ] ]. ], we obtain Notice what the transformation did to the input vector. It appears that the standard matrix rotated the input vector by 9 to yield the output vector. Is this a coincidence? Let s plug in a different vector and see what happens. If x =, then ] ( ) T = ]. The angle between the input vector and output vector is 9. Let s look at the standard matrix A and determine why it rotates any vector by 9. Note that cos(9 ) sin(9 ) A = = sin(9 ) cos(9 ) Should we be convinced that if we wish to rotate any vector by an angle θ that the matrix cos θ sin θ sin θ cos θ will do just that? Let s verify this. To find equations relating x and b = T (x), let φ be the angle from the positive x -axis to x, and let r be the common length of x and b (See Figure 3). Then from basic trigonometry, we have and x = r cos φ, x 2 = r sin φ () b = r cos(θ + φ), b 2 = r sin(θ + φ) (2)

9 Figure 3: Geometric effect of the rotation operator on R 2. Using trigonometric identities on (2) yields Then from the equations in (), we have b = r cos θ cos φ r sin θ sin φ b 2 = r sin θ cos φ + r cos θ sin φ b = x cos θ x 2 sin θ b 2 = x sin θ + x 2 cos θ. (3) which we can express in matrix-vector form as cos θ sin θ x = sin θ cos θ Note that the equations in (3) are linear, so T is a linear transformation. Furthermore, it follows from these equations that the standard matrix for T is cos θ sin θ A =. sin θ cos θ We have now established the fact that any vector in R 2 is rotated by an angle θ whenever the rotation matrix A acts on it by matrix-vector multiplication. x 2 b b 2 ].