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Mark each statement True or False. Justify each answer on the basis of a careful reading of the text.a. Two vectors are linearly dependent if and only if they lie on a line through the origin.b. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.c. If $\mathbf{x}$ and $\mathbf{y}$ are linearly independent, and if $\mathbf{z}$ is in Span $\{\mathbf{x}, \mathbf{y}\},$ then $\{\mathbf{x}, \mathbf{y}, \mathbf{z}\}$ is linearly dependent.d. If a set in $\mathbb{R}^{n}$ is linearly dependent, then the set contains more vectors than there are entries in each vector.

a. Trueb. Falsec. Trued. False

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 7

Linear Independence

Introduction to Matrices

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

03:12

$\mathcal{B}$ and $\mathca…

04:11

Mark each statement True o…

02:07

In Exercises 21 and 22, ma…

09:01

In Exercises 21 and $22,$ …

05:09

In Exercises 19 and $20,$ …

01:35

In Exercises 17 and $18,$ …

14:16

06:33

In Exercises 29 and $30, V…

06:10

In Exercises 19 and 20, V …

03:53

in this video, we're gonna be solving problem number 22 of section 1.7, which is a true falls four part question. I didn't wantto right all the words on the screen, So if you have the book in front of you, please follow on. So ace us if, ah, to back a states that two vectors air linearly dependent if and only if they lie on a line through the origin. And, um, that is true, because when a set of vectors air linearly independent, that means they're not linear combinations of each other. So they're gonna be, um, in separate planes, and they're gonna intersect the origin at different times, and they won't intersect it in a line. But if a set of vectors are literally dependent, then they're gonna be, ah, linear combination of each other. So they should, uh, they, like, extended within the same lines. So they passed through the origin of the same point. Simple enough for more. This is the exact same figure as Figure one on page 59 of the book. So it check it out for to make it easier, which is a zit looks cleaner in the book anyway, uh, the states that if a set of if a set of vectors contains fewer vectors than there are entries in the vectors, um, then the set is linearly independent. So they're made space that, um, saythe opposite of this where it says if you have more vectors, Ah, number of vectors is greater than a number of entries of the victors, then the set is linearly dependent, not literally independent, because that would result in free variables which had caused winner of the near dependency. So the opposite of this is not always not always true, because, um, just cause a set contains fewer vectors than their entries does not mean it is always linearly independent, and they're always examples to disprove it. Ah, see, states that if x and wire literally independent and if Z is in the span of X and y the net, then the set X y Z is linearly dependent. This is true, because if V three is in the span of the one and uh V to, they're gonna be in the same plane, Um, so they're gonna be literally dependent. And if we three is not in the span of you want me to. That means it's in its own plane. And if you draw the planet something like that, then it is literally independent that then the set is literally independent. Because, uh, no set of values would, uh, make the X equals zero equipped equation be just have the trivial solution. And finally, d is if a set is in rn. If a set in Oran is linearly dependent than set, contains more records than their entries in the In the Vector, it is not always, always true. As here, there's a counter example are, ah, the one of you to have the same entries. And there's two. So there's two entries into vectors, so there's not always more vectors than their entries. So here the number of actors and entries are equal. So these are linearly and dependent as one. Becker is a linear combination of the other as V two equals to be one. So by here, um, um Bye, dear. Uh, but he runs real quick by the time seven, um, in the book, this set is linearly dependent. So? So Stephen B is false

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