Chapter 3

1. Chapter 3 Section 3.5 Dimension

2. Dimension of a Subspace The vectors that make up a basis for a subspace W are not uniquely determined. That is, there might be more than one way to find a set of basis vectors for a subspace. For example multiplying a set of basis vectors by non zero scalars produces another basis. The sets and can both be a basis for the subspace W and not have any vectors in common. There is a feature these two sets will always have in common and that is the number of vectors that are required to make up a basis. In other words if the sets and are both a basis for the subspace Wthen (i.e. the sets will have the same number of vectors. The number of vectors required to form a basis for a subspace W is called the dimension of the subspace W. We denote this by . If is a basis for a subspace W then . The dimension of a subspace gives a way to recognize familiar geometric objects even though they might exist in more dimensions than can be physically demonstrated. It is a way to recognize when certain shapes are the same or different or part of another shape in dimensions beyond three. A 1-dimensional subspace is a line. A 2-dimensional subspace is a plane. A 3-dimensional subspace is space. A 4-dimensional subspace is hyperspace. (Or what physicists call space-time.)

3. Dimension of Since the standard basis for consists of the vectors the dimension of is n or we say that is n-dimensional. Example The subspace W described to the right is a "line" even though it exists in 5-dimensional space. The subspace W is 1-dimensional since it is the span of a single nonzero vector. Example The subspace W of that is the span of the 2 linearly independent vectors below is a 2-dimensional subspace of a 3-dimensional space. z y Properties of Dimension Let W be a subspace of with then each of the following to the right is always true. Any set of or more vectors in W is linearly dependent. Any set of fewer than vectors does not span W. Any set of linearly independent vectors is a basis of W. Any set of vectors that span W is a basis for W. x

4. Dimensions of Subspaces of Matrices Let A be an matrix. We know that the null space of A (i.e. ) and the range of A (i.e. ) are subspaces and each will have a dimension. The dimension of the null spaces is called the nullity of A and the dimension of the range is called the rank of A. Alternate Characterizations of Nullity and Rank of a Matrix We have seen the equivalence of certain subspaces of matrix A such as the range of A is the same subspace as the column space of A. A basis for the is found by taking the corresponding columns of A as the columns with leading 1’s in reduced echelon form. A basis for is found by taking the vectors corresponding to each independent variable in the vector form of the solution. We get the following equivalences: = dim(column space A) = number of leading 1’s = dim(row space of A) = number of dependent variables = number of independent variables This gives a very famous relationship between the rank and the nullity of a matrix A. This is a consequence of each variable either being dependent or independent and each column of the matrix A represents a variable.

5. Example For the matrix A to the right find: , , dim(row space of A), dim(column space A), , First for the matrix A we row reduce the matrix. Count the number of leading 1’s to get the rank. , , dim(column space A)=3, dim(row space A) = 3 Row reduced For the matrix form the matrix and again row reduce it. , Row reduced Relations Between A and Since the row space of a matrix A is equal to the column space of the matrix we get a couple very obvious relations one that is not so obvious is that the is always equal to the . Notice this is not true for the nullity these can be different values!

6. Matrix Properties Concerning Rank Many properties of matrices can be characterized in terms of rank. A system of equations, , is consistent if and only if : A matrix A is nonsingular if and only if .