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In Exercises 17 and 18 , $A$ is an $m \times n$ matrix. Mark each statement True or False. Justify each answer.a. If $B$ is any echelon form of $A$ , then the pivot columns of $B$ form a basis for the column space of $A$ .b. Row operations preserve the linear dependence relations among the rows of $A$ .c. The dimension of the null space of $A$ is the number of columns of $A$ that are not pivot columns.d. The row space of $A^{T}$ is the same as the column space of $A .$e. If $A$ and $B$ are row equivalent, then their row spaces are the same.

A. falseB. falseC. trueD. trueE. true

Calculus 3

Chapter 4

Vector Spaces

Section 6

Rank

Vectors

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problem problem. 18. We start with part. Hey, it says, um if bees any edge inform off A than the people call himself, it be former basis for the column. Space of A. So this Damon actually is false. The reason is that the column space the basis optimum off the com space off A is actually not people Columns A B is the corresponding columns in a or his funding columns in a. So, actually, we're taking the columns of A instead of the national home. Okay. Okay. So the next R B, we have Roe operations preserved the win your dependence relations about the rose off, eh? When it isn't stroke, all's well didn't steal false. Why? Well, because we we've performed the, uh We performed the road operations we interchange. You didn't inter change in the rose. Then everything will be messed up. Interchanging, rose, mess up things right? Because we are interchangeable. Rosen. We cannot tell which Rosie. It's exactly the the linear dependence relations we want we're looking for Okay. Hard to see so exas. The dimension of the non space of a is the number of columns of a that are not people. Columns Well, this this statement is true. You guess by our wrecked the room we have mentioned up in all space off a is and minus drank a So if we know the dimension of the no space of a number of the columns off, eh? There are not people columns, but rank of A actually keeps our the number of people columns. So those are not people. Columns is actually the columns that the total comes minus the people call him. So this is a you correctly that not people columns. All right. So hearty, it says means the rose space up. Hey, transposed the same as a common space of a Well, this is also true because that's how we perform the transpose operation. Just a bite transpose operation. Okay, Because by the transpose operation, the rose becomes the columns, columns becomes the rose. So the the rose face the game will be the same as a con space. When you do the transpose. Okay, our e So it says if a and B are really equivalent, then they're roast bases are the same. But it's also true. The reason is that you know the what we have to really prevalent people in the matrices that it's actually how we do how we find tree, how we find Rose face so we don't

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