T or F
1) Any N vectors spanning R^n are linearly independent
2)R5 has 7 linearly independent vectors
3) If a set of vectors with n elements is linearly dependent,
then a set with n - 1 elements is also linearly dependent
4) There exists a Linear Function T:R^n -> R^n such that the
range and the kernel of T are equal.
5) If a vector space has a dimension of n, then a basis for the
vector space will...
For each family of vectors, determine wether the vectors are
linearly independent or not, and in case they are linearly
dependent, find a linear relation between them.
a) x1 = (2, 2, 0), x2 = (0, 2, 2), x3 = (1, 0, 1)
b) x1 = (2, 1, 0), x2 = (0, 1, 0), x3 = (1, 2, 0)
c) x1 = (1, 1, 0, 0), x2 = (0, 1, 1, 0), x3 = (0, 0, 1, 1), x4 =...
Let ?1=(1,0,1,0) ?2=(0,−1,1,−1) ?3=(1,1,1,1) be linearly
independent vectors in ℝ4.
a. Apply the Gram-Schmidt algorithm to orthonormalise the vectors
{?1,?2,?3} of vectors {?1,?2,?3}.
b. Find a vector ?4 such that {?1,?2,?3,?4} is an orthonormal basis
for ℝ4 (where ℝ4 is the Euclidean space, that is, the
inner product is the dot product).
Determine whether the members of the given set of vectors are
linearly independent. If they are linearly dependent, find a linear
relation among them of the form c1x(1) + c2x(2) + c3x(3) = 0. (Give
c1, c2, and c3 as real numbers. If the vectors are linearly
independent, enter INDEPENDENT.) x(1) = 9 1 0 , x(2) = 0 1 0 , x(3)
= −1 9 0
There are three vectors in R4 that are linearly independent but
not orthogonal: u = (3, -1, 2, 4), v = (-2, 7, 3, 1), and w = (-3,
2, 4, 11). Let W = span {u, v, w}. In addition, vector b = (2, 1,
5, 4) is not in the span of the vectors. Compute the orthogonal
projection bˆ of b onto the subspace W in two ways: (1) using the
basis {u, v, w} for W, and...
Show that, in n-dimensional space, any n + 1 vectors are
linearly dependent.
HINT: Given n+1 vectors, where each vector has n components,
write out the equations that determine whether these vectors are
linearly dependent or not. Show that these equations constitute a
system of n linear homogeneous equations with n + 1 unknowns. What
do you know about the possible solutions to such a system of
equations?
Determine whether each set of vectors is linearly dependent or
linearly independent.
a) (1,1,0,1), (1,0,1,1), (0,1,1,1)
b) (1,0,1,0), (0,1,0,1), (1,-1,1,-1), (1,-1,0,0)