Question

Let (u,v,w,t) be a linearly independent list of vectors in R4. Determine if (u, v-u, w+5v, t) is a linearly independent list. Explain your reasoning and Show work.

Answer #1

Let u, vand w be linearly dependent vectors in a vector space V.
Prove that for any vector z in V whatsoever, the vectors u, v, w
and z are linearly dependent.

Prove that the span of three linearly independent vectors, u, v,
w is R3

Let T be a linear transformation that is one-to-one, and u, v be
two vectors that are linearly independent. Is it true that the
image vectors T(u), T(v) are linearly independent? Explain why or
why not.

Let S={u,v,w}S={u,v,w} be a linearly independent set in a vector
space V. Prove that the set S′={3u−w,v+w,−2w}S′={3u−w,v+w,−2w} is
also a linearly independent set in V.

vectors u=(1,2,3), v=(2,5,7), w=(1,3,5) are linearly dependent
or independent? (using echelon form)

Let T be a 1-1 linear transformation from a vector space V to a
vector space W. If the vectors u,
v and w are linearly independent
in V, prove that T(u), T(v),
T(w) are linearly independent in W

3-vectors u, v, and w satisfy u⋅(v ×w)=7. Find [u,v,w]⋅[v×w,
u×w,u×v]^T using properties of the triple scalar product.

determine the span of u=(1,2,0) v=(3,2,-1) w=(-2,0,1) and determine
if u,v, and w are linearly dependent.

Let u, v, and w be vectors in Rn. Determine which of the
following statements are always true. (i) If ||u|| = 4, ||v|| = 5,
and ?||u + v|| = 8, then u?·?v = 4. (ii) If ||u|| = 2 and ||v|| =
3, ?then |u?·?v| ? 5. (iii) The expression (v?·?w)u is both
meaningful and defined. (A) (ii) and (iii) only (B) (ii) only (C)
none of them (D) all of them (E) (i) only (F) (i) and...

Let W be the subspace of R4 spanned by the vectors a = 3e1 − 4e2
and b = e2 + e3 + e4. Find the orthogonal projection of the vector
v = [2, 0, 1, 0] onto W. Then calculate the distance of the point v
from the subspace W.

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