Math 18 Final Review

The set span{u, v} is always visualized as a plane through the origin.

• True

• False

Answer explanation

In most cases, yes, but not if v is a multiple of u or when one of u, v is the zero vector.

What is the characteristic polynomial of A and its corresponding eigenvalues?

W = {(4a+3b, 0, a+b+c, c-2a) : a,b,c are real numbers}

Is W a subspace? Justify your reasoning.

• Yes, and I have concrete justification

• No, and I know exactly why

• Yes, but I don't know how to prove/explain why

• No, but I am not sure why I think now

Answer explanation

The conditions that must be satisfied are ____. For a quicker solution, _____ can be found instead.

Let U be a subspace of R⁵.

Is dim(U) ≤  5.

• Yes, it must be exactly 5

• No, dimensions have nothing to do with R^5

• Yes, because of the properties of what dimensions refer to and I can explain.

• No since dimensions of a subspace must be larger than the number of elements in its constituent vectors so it fits

Answer explanation

A Basis has only ______ vectors (property) and there can be only ______ (number) such vectors. It is a subspace so it does not necessarily need to _______ the entire ______.

Is the set {(3,-1,2), (1,-1,-2)} orthogonal? Is it orthonormal?

• It is not orthogonal and not orthonormal.

• It is orthogonal and orthonormal.

• It is not orthogonal, but is orthonormal.

• It is orthogonal, but not orthonormal.

p

• p

• p

Determine whether the given transformation, T, is a linear transformation. If yes, prove it. Otherwise come up with a counter-example.

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OFF

Answer explanation

This is NOT a linear Transformation. You can show that as follows

Evaluate responses using AI:

OFF

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Evaluate responses using AI:

OFF

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Evaluate responses using AI:

OFF

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