1.1 Which vectors are linear combinations of v = (3,1) and w = (4,3) ?
1.2 Compare the dot product of v = (3,1) and w = (4,3) to the product of their lengths. Which is larger ? Whose inequality ?
1.3 What is the cosine of the angle between v and w in question 1.2 ? What is the cosine of the angle between the x - axis and v ?
2.1 Multiplying a matrix A times the column vector x = (2,-1) gives what combination of the columns of A ? How many rows and columns in A ?
2.2 If Ax = b then the vector b is a linear combination of what vectors from the matrix A ? In vector space language, b lies in the ____ space of A.
2.3 If A is the 2 by 2 matrix [2 1;6 6] what are its pivots ?
2.4 If A is the matrix [0 1;1 1] how does elimination proceed ? What permutation matrix P is involved ?
2.5 If A is the matrix [2 1;6 3] find b and c so that Ax = b has no solution and Ax = c has a solution.
2.6 What 3 by 3 matrix L adds 5 times row 2 to row 3 and then adds 2 times row 1 to row 2, when it multiplies a matrix with three rows ?
2.7 What 3 by 3 matrix E subtracts 2 times row 1 from row 2 and then subtracts 5 times row 2 from row 3 ? How is E related to L in question 2.6 ?
2.8 If A is 4 by 3 and B is 3 by 7, how many 'row times column' products go into AB ? How many 'column times row' products go into AB ? How many separate small multiplications are involved (the same for both) ?
2.9 Suppose A = [I U; 0 I] is a matrix with 2 by 2 blocks. What is the inverse matrix ?
2.10 How can you find the inverse of A by working with [A I] ? If you solve the n equations Ax = columns of I then the solutions x are columns of ____.
2.11 How does elimination decide whether a square matrix A is invertible ?
2.12 Suppose elimination takes A to U (upper triangular) by row operations with the multipliers in L (lower triangular). Why does the last row of A agree with the last row of L times U ?
2.13 What is the factorization (from elimination with possible row exchanges) of any square invertible matrix ?
2.14 What is the transpose of the inverse of AB ?
2.15 How do you know that the inverse of a permutation matrix is a permutation matrix ? How is it related to the transpose ?
3.1 What is the column space of an invertible n by n matrix ? What is the nullspace of that matrix ?
3.2 If every column of A is a multiple of the first column, what is the column space of A ?
3.3 What are the two requirements for a set of vectors in R^n to be a subspace ?
3.4 If the row reduced form R of a matrix A begins with a row of ones, how do you know that the other rows of R are zero and what is the nullspace ?
3.5 Suppose the nullspace of A contains only the zero vector. What can you say about solutions to Ax = b ?
3.6 From the row reduced form R, how would you decide the rank of A ?
3.7 Suppose column 4 of A is the sum of columns 1, 2, and 3. Find a vector in the nullspace.
3.8 Describe in words the complete solution to a linear system Ax = b.
3.9 If Ax = b has exactly one solution for every b, what can you say about A ?
3.10 Give an example of vectors that span R^2 but are not a basis for R^2.
3.11 What is the dimension of the space of 4 by 4 symmetric matrices ?
3.12 Describe the meaning of 'basis' and 'dimension' of a vector space.
3.13 Why is every row of A perpendicular to every vector in the nullspace ?
3.14 How do you know that a column u times a row v (both nonzero) has rank 1 ?
3.15 What are the dimensions of the four fundamental subspaces, if A is 6 by 3 with rank 2 ?
3.16 What is the row reduced form R of a 3 by 4 matrix of all 2's ?
3.17 Describe a 'pivot column' of A.
3.18 True ? The vectors in the left nullspace of A have the form A^T y.
3.19 Why do the columns of every invertible matrix yield a basis ?