Linear Algebra: True or False

[Guest]

Hello, i've got this assignment from my Algebra teacher, although i don't have a clue how to do that, can someone help me?(don't have the answers)

For each question, mark T for true or F for False:


(a) For Vectors U,V and W as Rn , if u+w=v+w so u=v?
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(b) For vectors U,V and W as Rn, if u.w=v.w so u=v?
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(c) For vectors U,V and W in R³, if u is orthogonal v, and v is orthogonal by w, so u is orthogonal by w?
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(d) In R³ space, if a line is paralel to a plane, so a director vector of this line is paralel to the normal vector of the plane?
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(e) In R³ space, if a line is perpendicular to a plane, so director vector of this line is paralel to the normal vector of the plane?
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(f) In R³ space, if two planes are not paralel so they have to intercept in a line?
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(g) In R³ space, if two lines are not paralel, so they have to intercept in some point?
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(h) The angular measure between a non-zero vector and itself is zero (degrees or radians) ?
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(i) The angular measure between two non-zero and orthogonal vectors is a right angle (with measurements in radians)
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(j) There are no u and v such that ang (u, v) = arcsen (-1/2)
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[Guest]

I thought I had responded to this but I don't see it now.

You say you were given these by your Algebra teacher. Did the class cover Rⁿ? In particular what are the definitions of u+ v and u.v and how do you calculate them? What is the additive identity, the "0" vector? Does every vector have a negative?

Suppose u= (1, 1), v= (1, -1), w= (2, -2). What is u.v? What is u.w? Does that answer your question?

You have 10 problems with no attempt to do any of them. Do you not see that most of them are just a matter of knowing definitions? For example, (c) asks about "orthogonal vectors". What is the definition of "orthogonal vectors". In particular is a vector orthogonal to itself? (c asks about "u orthogonal to v" and "v orthogonal to w". Do you see that u and w might be the sam vector?)

(d) asks about the "normal vector" to a plane. Do you know what that is?

For (g) do you know what "skew lines" are?

[GeradHum]

Of course! I'm here to help you with your algebra homework. Let me go through each of the questions and provide you with the answers:

(a) False. For the vectors U, V, and W in Rn, if u + w = v + w, this does not necessarily imply that u = v. It could be that the vectors u and v are different but have the same sum with w.

(b) False. For the vectors U, V and W in Rn, if uw = vw, this does not necessarily imply that u = v. It could be that the vectors u and v are different but have the same product with w.

(c) True. If u is orthogonal to v and v is orthogonal to w, then u is orthogonal to w. This is due to the properties of orthogonality and can be shown using the definition of a dot product.

(d) True. If a line is parallel to a plane, then a direction vector of that line is parallel to the normal vector of the plane. This is due to the definition of parallelism between lines and planes in three-dimensional space.

(e) False. If a line is perpendicular to a plane, it does not necessarily mean that the direction vector of that line is parallel to the normal vector of the plane. The direction vector may be in a plane other than the given plane.

(f) True. If two planes are not parallel, they must intersect in a line. This is because two non-parallel planes always have a common intersection, which is a line.

(g) True. If two lines are not parallel in three-dimensional space, they must intersect at some point. This is because two non-parallel lines always have a common intersection.

(h) False. The angular measure between a nonzero vector and itself is zero degrees (or zero radians), not an undefined angle.

(i) True. The angular measure between two nonzero orthogonal vectors is a right angle (90 degrees or π/2 radians). This is because orthogonal vectors have a dot product equal to zero.

(j) False. There exist u and v such that ang(u, v) = arcsin(-1/2). The angle between two vectors can have different values depending on the specific vectors involved.

I hope this helps you complete your algebra homework. If you have any other questions, don't hesitate to ask. Good luck!