Math 18 Final Review SP23

Let u, v, w be three linearly independent vectors in a vector space. Are the vectors x = u+v , y = v+w, z = u+w also linearly independent? Why?

• Yes, because the determinant of x, y, and z is nonzero

• No, because y is a linear combination of x and z

• No, for a different reason

• Yes, for a different reason

Answer explanation

 Let T be the reflection in the line y = x in R^2. So, T(x,y) = (y,x). Write down the standard matrix of T, and then use it to compute T(3, 4)

Answer explanation

Given P, Find the P^-1

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if the columns of A are linearly dependent, then det(A) = 0

• True

• False

Is the matrix U is an orthogonal matrix?

• Yes because all the vectors are orthogonal to each other

• Yes because all the vectors are orthogonal to each other and each vector has length 1

• No because the vectors are not all orthogonal to each other

• No because the dot product of the first vector with the second vector = 0

Find the vectors creating the eigenspace of λ=2

• (0 −1 1)^T

• (-1 1 0)^T & (0 0 1)^T

• (0 1 0)^T & (−1 0 1)^T

• (1 0 1)^T

Let U be a subspace of R⁵.

Is dim(U) ≤  5.

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

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

• Yes, it must be exactly 5

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

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 ______.

What is the basis of Nul(A)?

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What is the basis for Col(A)? (All vectors are transposed)

• span{[0 1 2], [2 -1 4]}

• {[2 0 0], [-1 1 0], [0 -3 0]}

• {[1 0 0], [0 1 0]}

• {[0 1 2], [2 -1 4]}