I\'m taking a linear algebra course and I haven\'t needed to doproofs before thi
ID: 2939158 • Letter: I
Question
I'm taking a linear algebra course and I haven't needed to doproofs before this class (I'm an EE major and we take "special"math for Calc and Diff Eq). I could really use some help on theproofs for vector spaces. I have a quiz on Tuesday and one of theseproofs will be on there.1. Let A and B be row equivalent matrices. (a) Show that the dimension of thecolumn space of A equals the dimension of the column space ofB. (b) Are the column spaces of thetwo matrices necessarily the same? Justify your answer.
2. Let A be an m x n matrix with rank equal ton. Show that if x 0, and y=Ax, then y 0.
3. Let A be an m x n matrix. (a) If B is a nonsingular m xn matrix, show that BA and A have the same nullspace and hencethe same rank. (b) If C is a nonsingular n xn matrix, show that AC and A have the same rank.
4. Let A Rm x n, B Rn xr, and C = AB. Show that (a) The column space of C is asubspace of the column space of A. (b) The row space of C is asubspace of the row space of B. (c) rank(C) min(rank(A),rank(B)).
1. Let A and B be row equivalent matrices. (a) Show that the dimension of thecolumn space of A equals the dimension of the column space ofB. (b) Are the column spaces of thetwo matrices necessarily the same? Justify your answer.
2. Let A be an m x n matrix with rank equal ton. Show that if x 0, and y=Ax, then y 0.
3. Let A be an m x n matrix. (a) If B is a nonsingular m xn matrix, show that BA and A have the same nullspace and hencethe same rank. (b) If C is a nonsingular n xn matrix, show that AC and A have the same rank.
4. Let A Rm x n, B Rn xr, and C = AB. Show that (a) The column space of C is asubspace of the column space of A. (b) The row space of C is asubspace of the row space of B. (c) rank(C) min(rank(A),rank(B)).
Explanation / Answer
Question 1 If A,B are row equivalent Then C A = B where C is the matrix containing the row operations that take us from A to B. So far so good. Now C is a matrix which rearranges the rows of A--hence the name row operations. We have operations of the sort: 3 * row1 - row 2. Thus the rows of B are just rearrangements of the rows of A. But rearrangements don't change the row space; hence the rank is the same; and since the dimension of the column space is the rank, the dimension of the two column spaces are the same. FormaLly, A row in the system will be [row 1 of C] [row 1 of A] = [row 1 of B]. And [row 1 of B] = [row 1 of A + c2 * row 2 of A + ... cm * row m of A] Note there's no coefficient for row 1 of A, thus if there's a pivot in that row in A there will be one also in B. Same #of pivots = same rank = same col space dimension! As for part b, the answer is of course no. A and B may have the same col space dimensions, but A is not B! Its columns are different and thus they will span different col spaces! Cheers, if I have time ill tackle 2 and 3, 4 I'm having a hard time understanding.