Problem 1(a). You are looking for one or more equations satisfied by vectors in the range of A. For a vector b to be in the range is the same as saying the equation AX = b is consistent. This problem is like example 4 on pp. 182-3. For the matrix A in that case an acceptable answer would be:
The range of A (the one in the textbook) is the set of vectors b satisfying the equation -3b1 + b2 + b3 = 0
So reduce the augmented matrix A|b in this way.
How to check: The column vectors of A should satisfy the equation!
Problem 1(b) Ignore this for now. We don't yet have the word 'basis'.
Problem 2ab. Use the method in pp.95-98 You can check your answer by computing CB. This should be I.
Problem 2c.
One solution: If MNP = I, then (MNP)P-1 = IP-1 = P-1. But (MNP)P-1 = MN. So MN = P-1.
Then N = M-1MN = M-1P-1.
So then N-1 = (P-1)-1(M-1)-1 = PM.
Alternate solution: If MNP = I, then (MNP)-1 = I-1= I. But I = (MNP)-1 = P-1 N-1 M-1.
So N-1 M-1 = P(P-1 N-1 M-1) = PI = P and N-1 = (N-1 M-1)M =PM.
Either way, N-1 = PM.
Problem 3a
Free variables are x1, x3, x6. And x2 = -2x3 – 3x6; x4 = -4x6; x5 = -5x6.
So your answer in vector form should look like X = x1W1 + x3W2 + x6W3. where the Wi are column vectors in R6.
W1 = [1 0 0 0 0 0]T
W2 = [0 -2 1 0 0 0]T
W3 = [0 -3 0 -4 -5 1]T
Problem 3b – ignore basis question
Problem 4a: see definition in text. Note that you need a complete sentence and then it is a set of vectors – not a matrix – that is linearly independent or dependent.
Problem 4b: this like the one I did in class Wednesday. One way is to convert the question to one about columns; take the transpose of G and check whether or not the columns of the transpose are linearly independent. A second way is to row-reduce A. If the echelon form has a zero row, then the row vectors are linearly dependent. And if no zero row, they are not.
Problem 4c: We know the columns have to be linearly dependent without doing any computation. See Theorem 11 page 76 and the previous example on pp. 75-6
NOTES ON TRUE-FALSE PROBLEMS
My answers are a lot more detailed that what you would have to give. You need to indicate that you have a clear reason for your answer. My answers are longer because I am trying to explain to you why the answer is what it is when you may not know the answer. Different audience.
Problem 5a: This is (very) false. If w is a non-zero vector, then the set {w} is linearly independent with k = 1. Or if this seems special, take any linearly independent set of two vectors in 3-space from your homework. What is tricky about this problem is that it resembles the statement of Theorem 11, so if oneีs concepts are hazy, one might leap the wrong way
Problem 5b: True. Did this in class. If we number so that v1 = 0, then the linear combination 1v1 + 0v2 + . . . 0vk = 0 is a linear combination whose coefficients are not all zero, so the set is linearly dependent by definition.
Problem 5c: If the range of A is Rn, then this says that Ax = b is consistent for any b, which means that the reduced row echelon form of A has no zero row, which (since A has n rows and n columns) that the reduced row echelon form of A = I. This means that the only solution of AX = 0 is X = 0, which means the columns of A are linearly independent. This means that A is a non-singular matrix (by definition).
Note: This condition on the range was one of the about 5 properties of non-singular matrices that were listed in class one day.