4.1 What does the word 'complement' mean about orthogonal subspaces ?
4.2 If V is a subspace of the 7-dimensional space R^7, the dimensions of V and its orthogonal complement add to ____ .
4.3 The projection of b onto the line through a is the vector _____ .
4.4 The projection matrix onto the line through a is P = _____ .
4.5 The key equation to project b onto the column space of A is the 'normal equation' _____ .
4.6 The matrix A^T A is invertible when the columns of A are _____ .
4.7 The least squares solution to Ax = b minimizes what error function ?
4.8 What is the connection between the least squares solution of Ax = b and the idea of projection onto the column space ?
4.9 If you graph the best straight line to a set of 10 data points, what shape is the matrix A and where does the projection p appear in the graph ?
4.10 If the columns of Q are orthonormal, why is Q^T Q = I ?
4.11 What is the projection matrix P onto the columns of Q ?
4.12 If Gram-Schmidt starts with the vectors a = (2,0) and b = (1,1), which two orthonormal vectors does it produce ? If we keep a = (2,0)does Gram-Schmidt always produce the same two orthonormal vectors ?
4.13 True ? Every permutation matrix is an orthogonal matrix.
4.14 The inverse of the orthogonal matrix Q is _____ .
5.1 What is the determinant of the matrix -I ?
5.2 Explain how the determinant is a linear function of the first row.
5.3 How do you know that det inv(A) = 1/ det A ?
5.4 If the pivots of A (with no row exchanges) are 2, 6, 6, what submatrices of A have known determinants ?
5.5 Suppose the first row of A is 0, 0, 0, 3. What does the "big formula" for the determinant of A reduce to in this case ?
5.6 Is the ordering (2,5,3,4,1) even or odd ? What permutation matrix has what determinant, from your answer ?
5.7 What is the cofactor C_23 in the 3 by 3 elimination matrix E that subtracts 4 times row 1 from row 2 ? What entry of inv(E) is revealed ?
5.8 Explain the meaning of the cofactor formula for det A using column 1.
5.9 How does Cramer's Rule give the first component in the solution to Ix = b ?
5.10 If I combine the entries in row 2 with the cofactors from row 1, why is a_21 C_11 + a_22 C_12 + a_23 C_13 automatically zero ?
5.11 What is the connection between determinants and volumes ?
5.12 Find the cross product of u = (0,0,1) and v = (0,1,0) and its direction.
5.13 If A is n by n, why is det(A - xI) a polynomial of degree n ?