Linear algebra review

Transcription

1 EE263 Autumn 2015 S. Boyd and S. Lall Linear algebra review vector space, subspaces independence, basis, dimension nullspace and range left and right invertibility 1

2 Vector spaces a vector space or linear space (over the reals) consists of a set V a vector sum + : V V V a scalar multiplication : R V V a distinguished element 0 V which satisfy a list of properties 2

3 Vector space axioms x + y = y + x, x, y V + is commutative (x + y) + z = x + (y + z), x, y, z V + is associative 0 + x = x, x V 0 is additive identity x V ( x) V s.t. x + ( x) = 0 existence of additive inverse (αβ)x = α(βx), α, β R x V scalar mult. is associative α(x + y) = αx + αy, α R x, y V right distributive rule (α + β)x = αx + βx, α, β R x V left distributive rule 1x = x, x V 1 is multiplicative identity 3

4 Examples V 1 = R n, with standard (componentwise) vector addition and scalar multiplication V 2 = {0} (where 0 R n ) V 3 = span(v 1, v 2,..., v k ) where and v 1,..., v k R n span(v 1, v 2,..., v k ) = {α 1v α k v k α i R} 4

5 Subspaces a subspace of a vector space is a subset of a vector space which is itself a vector space roughly speaking, a subspace is closed under vector addition and scalar multiplication examples V 1, V 2, V 3 above are subspaces of R n 5

6 Vector spaces of functions V 4 = {x : R + R n x is differentiable}, where vector sum is sum of functions: (x + z)(t) = x(t) + z(t) and scalar multiplication is defined by (a point in V 4 is a trajectory in R n ) (αx)(t) = αx(t) V 5 = {x V 4 ẋ = Ax} (points in V 5 are trajectories of the linear system ẋ = Ax) V 5 is a subspace of V 4 6

7 (Euclidean) norm for x R n we define the (Euclidean) norm as x = x x x2 n = x T x x measures length of vector (from origin) important properties: αx = α x x + y x + y x 0 x = 0 x = 0 homogeneity triangle inequality nonnegativity definiteness 7

8 RMS value and (Euclidean) distance root-mean-square (RMS) value of vector x R n : rms(x) = ( 1 n n i=1 x 2 i ) 1/2 = x n norm defines distance between vectors: dist(x, y) = x y x x y y 8

9 Independent set of vectors a set of vectors {v 1, v 2,..., v k } is independent if α 1v 1 + α 2v α k v k = 0 = α 1 = α 2 = = 0 some equivalent conditions: coefficients of α 1v 1 + α 2v α k v k are uniquely determined, i.e., α 1v 1 + α 2v α k v k = β 1v 1 + β 2v β k v k implies α 1 = β 1, α 2 = β 2,..., α k = β k no vector v i can be expressed as a linear combination of the other vectors v 1,..., v i 1, v i+1,..., v k 9

10 Basis and dimension set of vectors {v 1, v 2,..., v k } is called a basis for a vector space V if V = span(v 1, v 2,..., v k ) and {v 1, v 2,..., v k } is independent equivalently, every v V can be uniquely expressed as v = α 1v α k v k for a given vector space V, the number of vectors in any basis is the same number of vectors in any basis is called the dimension of V, denoted dim V 10

11 Nullspace of a matrix the nullspace of A R m n is defined as null(a) = { x R n Ax = 0 } null(a) is set of vectors mapped to zero by y = Ax null(a) is set of vectors orthogonal to all rows of A null(a) gives ambiguity in x given y = Ax: if y = Ax and z null(a), then y = A(x + z) conversely, if y = Ax and y = A x, then x = x + z for some z null(a) null(a) is also written N (A) 11

12 Zero nullspace A is called one-to-one if 0 is the only element of its nullspace null(a) = {0} Equivalently, x can always be uniquely determined from y = Ax (i.e., the linear transformation y = Ax doesn t lose information) mapping from x to Ax is one-to-one: different x s map to different y s columns of A are independent (hence, a basis for their span) A has a left inverse, i.e., there is a matrix B R n m s.t. BA = I A T A is invertible 12

13 Two interpretations of nullspace suppose z null(a), and y = Ax represents measurement of x z is undetectable from sensors get zero sensor readings x and x + z are indistinguishable from sensors: Ax = A(x + z) null(a) characterizes ambiguity in x from measurement y = Ax alternatively, if y = Ax represents output resulting from input x z is an input with no result x and x + z have same result null(a) characterizes freedom of input choice for given result 13

14 Left invertibility and estimation ˆx B y A x apply left-inverse B at output of A then estimate ˆx = BAx = x as desired non-unique: both B and C are left inverses of A 1 0 A = 0 1 B = 1 0 [ 1 0 ] C = [ ]

15 Range of a matrix the range of A R m n is defined as range(a) = {Ax x R n } R m range(a) can be interpreted as the set of vectors that can be hit by linear mapping y = Ax the span of columns of A the set of vectors y for which Ax = y has a solution range(a) is also written R(A) 15

16 Onto matrices A is called onto if range(a) = R m equivalently, Ax = y can be solved in x for any y columns of A span R m A has a right inverse, i.e., there is a matrix B R n m s.t. AB = I rows of A are independent null(a T ) = {0} AA T is invertible 16

17 Interpretations of range suppose v range(a),w range(a) y = Ax represents measurement of x y = v is a possible or consistent sensor signal y = w is impossible or inconsistent; sensors have failed or model is wrong y = Ax represents output resulting from input x v is a possible result or output w cannot be a result or output range(a) characterizes the possible results or achievable outputs 17

18 Right invertibility and control y A x C y des apply right-inverse C at input of A then output y = ACy des = y des as desired 18

19 Inverse A R n n is invertible or nonsingular if it has both a left and right inverse equivalent conditions: columns of A are a basis for R n rows of A are a basis for R n y = Ax has a unique solution x for every y R n null(a) = {0} range(a) = R n 19

20 Inverse if a matrix A has both a left inverse and a right inverse, then they are equal BA = I and AC = I = B = C hence if A is invertible then the inverse is unique AA 1 = A 1 A = I 20

21 Interpretations of inverse suppose A R n n has inverse B = A 1 mapping associated with B undoes mapping associated with A (applied either before or after!) x = By is a perfect (pre- or post-) equalizer for the channel y = Ax x = By is unique solution of Ax = y 21

22 Dual basis interpretation let a i be columns of A, and b T i be rows of B = A 1 from y = x 1a x na n and x i = b T i y, we get y = n ( b T i y)a i i=1 thus, inner product with rows of inverse matrix gives the coefficients in the expansion of a vector in the columns of the matrix { b 1,..., b n} and {a 1,..., a n} are called dual bases 22

23 Change of coordinates 0. standard basis vectors in R n : (e 1, e 2,..., e n) where e i = 1 (1 in ith component). 0 obviously we have x = x 1e 1 + x 2e x ne n x i are called the coordinates of x (in the standard basis) if (t 1, t 2,..., t n) is another basis for R n, we have x = x 1t 1 + x 2t x nt n where x i are the coordinates of x in the basis (t 1, t 2,..., t n) then x = T x and x = T 1 x 23

24 Similarity transformation consider linear transformation y = Ax, A R n n express y and x in terms of t 1, t 2..., t n, so x = T x and y = T ỹ, then ỹ = (T 1 AT ) x A T 1 AT is called similarity transformation similarity transformation by T expresses linear transformation y = Ax in coordinates t 1, t 2,..., t n 24