Chapter 3

1. Chapter 3 Section 3.3 Examples of Subspaces

2. Span of a Set of Vectors If is a set of vectors in then all of the vectors that are linear combinations of the vectors in S is called the span of S and is denoted Span of a Set of Vectors is a Subspace For any finite set of vectors in the set is a subspace of To show this let and . Show (a3): which means Show (c1): Let and then which means Show (c2): Let then Example The line to the right can be written as a span of a nonzero vector. In fact a line through the origin in higher dimensions is the span of a single nonzero vector.

3. Example The span of the two vectors to the right is a plane passing through the origin. We can find the equation of this plane in 2 ways. Method 1 (Geometric) Compute the normal vector using the cross product and plug into the equation for a plane. This plane has equation . This method will only work for 3 dimensions. • Method 2 (Algebraic) • Let the vector be a linear combination in the span. Set the linear combination equal to b and row reduce the matrix to echelon form. In order for this system to be consistent it can not have a nonzero number in the augmented column. This means: Setting we get: 2R1+R2 -R1+R3 -R2+R3

4. Example (Continued) In general a plane is the span of any two nonzero vectors that point in different directions in any dimension. This example showed that a subspace is always a solution to a homogeneous system of equations. The homogeneous system of equations that a subspace of vectors satisfies is called and algebraic specification of the subspace. Example Give an algebraic specification for the line given below. Now, Form the augmented matrix with the vector: 4R1+R2 -3R1+R3 Again to be consistent both of the last rows must be zero this gives two equations The intersection of two planes is a line. Both of the planes above contain the line , so this is the line of intersection.

5. Matrix Row & Column Vectors Let A be an matrix. Let be the row vector that is the row of matrix A. Let be the column vector that is the column of matrix A. Example Find the row and column vectors for the matrix B given to the right. • : : Row Space of a Matrix The row space of a matrix A is the span of the row vectors of A. This is a subspace since it is the span of a set of vectors. Row Space A = Row Space B

6. The Column Space of a Matrix The column space of a matrix A is the span of the column vectors of A. Again this is a subspace since the span of any set of vectors is a subspace Column Space A = If then Column Space B = The Null Space (Kernel) of a Matrix If A is a matrix and x is a vector in then is a vector in . The null space of A (denoted ) are all vectors such that (the zero vector) in . In a previous example we showed the kernel of a matrix was a subspace. Show (a3): which means Show (c1): If then and then which means 3. Show (c2): If and then , then which means The null space of a matrix A are the set of solutions to the homogeneous system of equations with coefficient matrix A. So every subspace is the null space of some matrix.

7. The Range of a Matrix If A is a matrix and x is a vector in then is a vector in . The range of A (denoted ) are all vectors such that for some vector xin . Because of the way matrix multiplication is defined (row  column) the matrix product can again be rewritten as a linear combination of the columns of the matrix. Column Space of A Since this is the case for any combination of the range of A will be the span of the columns of the matrix and we see that the range and the column space are equal. In particular this shows that the range is a subspace. This gives us several different ways to characterize a subspace, all of them in terms of a matrix. Not only are subspaces solutions of homogeneous systems of equations but they are various combinations of parts of matrices!