Monthly Archives: September 2011

Exercise 2.3.13.What are the dimensions of the following spaces? a) vectors in with components that sum to zero b) the nullspace associated with the 4 by 4 identity matrix c) the space of all 4 by 4 matrices Answer: a) … Continue reading →

Exercise 2.3.12.Suppose that the set of vectors , , ,  and ,  is a basis for and that is a subspace of . Provide a counterexample to the conjecture that some subset of , , ,  and is necessarily a … Continue reading →

Exercise 2.3.11. Consider the subspace of consisting of all vectors whose first two components are equal. Find two different bases for this subspace. Answer: All vectors in the subspace are of the form . One basis for the subspace consists … Continue reading →

Exercise 2.3.10. The set of all 2 by 2 matrices forms a vector space under the standard rules for multiplying two matrices and multiplying a matrix by a scalar. Find a basis for the space and describe the subspace spanned … Continue reading →

Exercise 2.3.9. Give a basis for the column space of the matrix and express the other columns of in terms of it. Find a matrix that is reduced by elimination to the same but has a different column space than … Continue reading →

Exercise 2.3.8. Describe the column space of the matrix and give a basis for it. Do the same for . Answer:The second column of is twice the first column, so that the two vectors are linearly dependent. The column space … Continue reading →

Exercise 2.3.7. For each of the following, state whether the vector is in the subspace spanned by . (Construct a matrix with as the columns, and try to solve .) a) , , , b) , , , , any … Continue reading →

Exercise 2.3.6. What are the geometric entities (e.g., line, plane, etc.) spanned by the following sets of vectors: a) , , and b) , , and c) the combined set of six vectors above (which vectors are a basis for … Continue reading →

Exercise 2.3.5. If we test vectors by independence by putting them into the rows of a matrix (rather than the columns), how can we determine whether or not the rows are independent using the elimination process? Determine the independence of … Continue reading →

Exercise 2.3.4. Suppose the vectors , , and are linearly independent. (The book says linearly dependent, but I believe this is a typo.) Are the vectors , , and also linearly independent? Answer: Consider the linear combination of , , … Continue reading →