(1) For each of the following false statements give a counter-example (i.e. an example set of vectors which demonstrate that the statement is false). You may, but do not have to, reuse your example vectors. (a) If 7 and w are any two vectors then || 7 – w|| = || 7|| – || 1||- %3D (b) If 7 and w are any two non-parallel three dimensional vectors then there is only one unit vector perpendicular to both 7 and w. (c) If 7 and w are any two three dimensional vectors then 7 x w + w × d.

Suppose a₁, a2, a3, a4, and a5 are vectors in R ³, A = (a₁ | 8₂ | a3 | a | a5), and a. Select all of the true statements (there may be more than one correct answer). 3 A. span{a₁, a2, a3, a4} = R ³ 3 | B. span{a₁, a2, a3} = R³ C. {a₁, a₂} is a linearly independent set D. {a₁, a₂ } is a basis for R ³ 3 | E. {a₁, A₂, A3, A4} is a basis for R F. {a₁, a₂, a3} is a linearly independent set G. {a₁, a₂, a3 } is a basis for R ³ 3 3 H. span{a₁, a₂} R ³ = rref(A) = |I. {A₁, A₂, A3, A4} is a linearly independent set 9 1 0 0 1 0 0 -4 0 -1 1 1 NL 2 2

7) Use coordinate vectors to determine whether the set H = { 1 + 2x + 3 x ²₁ x + 4x² ₂ 2 + 5x + x²} CP ₂ 2 is Inearly independent Fill in the Blank: The coordinate vectors can be P. 2 used here, because P₂ RO to 15 Isomorphic

§7. (6 p) Let each set S1, S2 be formed by only one vector: S₁ Is S₁ LI, LD or neither? Is S₂ LI, LD or neither? Is the set { , } LI, LD or neither? = { 7 }, S₂ = { 0

State whether the following statements are True or False. If they are true, provide mathematical reasonin are false, provide an example for which it is not true (a counterexample) or prove it is not true. a) (7 · î)² + ( · §)² + (& · k)² = |vl°. j)? + (5 . k)² = |vſ. Note: If you say the above statement is true, it is not sufficient to show it for a few numerical examples. to show it's true for an arbitrary vector 7. b) (i x j) and (i × (i × (i × j))) are parallel. c) x² + y? + 3z2 = 1 is the equation of a sphere. d) The value ủ :(ũ × w) is always zero.

If 1 and 2 are vectors in R³ and 7₁ 72, then 1 and 2 must be linearly independent. True O False