Independence, basis, and dimension What does it mean for vectors to be independent? How does the idea of inde pendence help us describe subspaces like the nullspace? Linear independence Suppose A is an m by n matrix with m < n (so Ax = b has more unknowns than equations). A has at least one free variable, so there are nonzero solutions to Ax = 0. A combination of the columns is zero, so the columns of this A are dependent. We say vectors x1 , x2 , ...xn are linearly independent (or just independent) if c1 x1 + c2 x2 + · · · + cn xn = 0 only when c1 , c2 , ..., cn are all 0. When those vectors are the columns of A, the only solution to Ax = 0 is x = 0. Two vectors are independent if they do not lie on the same line. Three vectors are independent if they do not lie in the same plane. Thinking of Ax as a linear combination of the column vectors of A, we see that the column vectors of A are independent exactly when the nullspace of A contains only the zero vector. If the columns of A are independent then all columns are pivot columns, the rank of A is n, and there are no free variables. If the columns of A are dependent then the rank of A is less than n and there are free variables. Spanning a space Vectors v1 , v2 , ...vk span a space when the space consists of all combinations of those vectors. For example, the column vectors of A span the column space of A. If vectors v1 , v2 , ...vk span a space S, then S is the smallest space containing those vectors. Basis and dimension A basis for a vector space is a sequence of vectors v1 , v2 , ...vd with two proper ties: • v1 , v2 , ...vd are independent • v1 , v2 , ...vd span the vector space. The basis of a space tells us everything we need to know about that space. Example: R3 �� One basis for R3 is � c1 1 0 0 � � 1 0 0 � , 0 1 0 � � , � + c2 0 1 0 0 0 1 �� . These are independent because: � + c3 0 0 1 � � = 0 0 0 � is only possible when c1 = c2 = c3 = 0. These vectors span R3 . � � � As discussed at the start of Lecture 10, the vectors 1 1 2 , 2 2 5 � � and 3 3 8 � do not form a basis for R3 because these are the column vectors of a matrix that has two identical rows. The three vectors are not linearly independent. In general, n vectors in Rn form a basis if they are the column vectors of an invertible matrix. Basis for a subspace � � The vectors 1 1 2 and � 2 2 5 � span a plane in R3 but they cannot form a basis for R3 . Given a space, every basis for that space has the same number of vec tors; that number is the dimension of the space. So there are exactly n vectors in every basis for Rn . Bases of a column space and nullspace Suppose: ⎡ ⎤ 1 2 3 1 A = ⎣ 1 1 2 1 ⎦. 1 2 3 1 By deﬁnition, the four column vectors of A span the column space of A. The third and fourth column vectors are dependent on the ﬁrst and second, and the ﬁrst two columns are independent. Therefore, the ﬁrst two column vectors are the pivot columns. They form a basis for the column space C ( A). The matrix has rank 2. In fact, for any matrix A we can say: rank( A) = number of pivot columns of A = dimension of C ( A). (Note that matrices have a rank but not a dimension. Subspaces have a dimen sion but not a rank.) The column vectors of this A are not independent, so the nullspace N ( A) contains more than just the zero vector. Because the third column is the sum −1 ⎢ −1 ⎥ of the ﬁrst two, we know that the vector ⎣ is in the nullspace. Similarly, 1 ⎦ 0 ⎡ ⎤ −1 ⎢ 0 ⎥ ⎣ 0 ⎦ is also in N ( A). These are the two special solutions to Ax = 0. We’ll −1 ⎡ ⎤ see that: dimension of N ( A) = number of free variables = n − r, so we know that the dimension of N ( A) is 4 − 2 = 2. These two special solu tions form a basis for the nullspace.