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Each statement in Exercises 33–38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.)If $\mathbf{v}_{1}, \ldots, \mathbf{v}_{4}$ are in $\mathbb{R}^{4}$ and $\mathbf{v}_{3}$ is not a linear combination of $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{4},$ then $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}, \mathbf{v}_{4}\right\}$ is linearly independent.

False. See explanation.

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 7

Linear Independence

Introduction to Matrices

Oregon State University

Harvey Mudd College

Idaho State University

Lectures

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in this video, we're gonna be solving question number 36 from section 1.7, which is based on the new Independence. So the book gives us a statement and asked us the prove it true for all possible cases O r prove it falls by using one counter example. And the same for Question 36 is, uh, given for Colin vectors and our four. And if v three is not a linear combination of V one v two and before than the set V one b two B three before it is literally independent. This is what the statements claiming. But, um, this is false as, um just cause just cause of the three is linearly independent doesn't guarantee uh, the set v one, V two, and before to be linearly independent either. Ah, for a simple counter example, we can choose. Ah, let's say V three equals 100 uh, for victory and be one, uh, we can choose 011 You too can d'oh! 055 And before you can be like is your 11 11 here. It is clear that, um, even though the three is linearly independent, as it cannot be formed with by any combination of V one, V two and before combined together. We know that V two is a linear combination of one as it is just five times one R 500 B one and before is just 11 times be one. So V two equals five b one. So sorry about the fives and the four goals 11 be born. So here, even though the one or be three is literally independent, the other three vectors air not linearly, independent of each other. So that makes the entire set. Do you want me to be there and before be linearly dependent, not linearly independent. So this statement is false.

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