Quiz 4 Ma 322 005 Name ______________

1. What is meant by the column space of a matrix A? Answer in a sentence (or two) and illustrate with the matrix ,

The column space of a matrix A is the set of linear combinations of the columns of A. Another way to say the same thing is: the column space of A is the set of all vectors b such that Ax=b for some vector x.

In the case of , the column space is seen to be all of . One way to see this is to solve the equation for s and t in terms of x and y, getting .

2. What is meant the null space of a matrix A? Answer in a sentence (or two) and illustrate with the matrix .

The null space of a matrix A is the set of solutions to the equation Ax = O. For the case , we could solve the equation Ax = O by writing out the equations x + 2*y = 0, x + 2*y + 3*z = 0 and solve for x and z in terms of y: x = -2*y, y = y, z=0. So each vector in N(A) is of the form . Thus the null space is the line of scalar multiples of .

Remark: Using the algorithm given by Prof. Sathaye in the supplemental lecture to chapter 3, the last parts of problems 1 and 2 can be done at once.

leads to and from this matrix we can read off (transposes of) bases for the column space and the null space of A (the 1st and 3 pivot rows of and the row augmenting the row of 0's in )

3. What is the smallest subspace of containing the plane x + y + z = 1? Explain your answer .

The smallest subspace containing this plane contains the vectors , therefore it contains any linear combination of them. But each vector in is a linear combination of these three vectors. So the answer is all of .