Show that there do not exist scalars c₁, c₂, and C3 such that c₁(1,0, 1,0) + c2 (1, 0, 2, 1) + c3(2,0, 1, 2) = (1, -2, 2, 3)

Find a basis for the plane 4x + 3y + 5z = 0 in R³. |

22. Show that there do not exist scalars c₁, c₂, and c² such that c₁(1,0, 1,0) + c₂(1, 0, −2, 1) + c3(2, 0, 1, 2) = (1, -2, 2, 3)

Construct two bijections from (0,1) to (1,1/0).

Suppose that ū, ī and w are vectors in a real inner product space such that (7, w) = 16, (T, w) = 5, d(2ū, v) = v27 and ||W|| = V27. Show that 27 - 0 - w = 0.

"On R², define the operations of addition and scalar multiplication as follows: (X₁, X₂) + (y₁ Y₂) = (x₂ + y₂ +1, x₂ + y₂ + 1), c(x₁, x₂) = (Cx₂ + C-1, ₂+c-1). Then R² with these operations forms a vector space." Note that here the zero vector is (-1,-1) and the additive inverse of (x1, x2) is (-X₁-2, -x₂-2), not (-x₂-x₂)- a) Show that (-1, -1) acts as the zero vector under these definitions.

(b) (c) 0 (69) (89-19) 0 0 0 0 (6 9.89.163) 0 0 0 0 2 1 0 0 2 4. Determine whether the following vectors are lin- early independent in 2×2: 0 (@) [9] [8

Find all scalars c₁, c₂ and c3 such that C2= ㅁ . eTextbook and Media W c₁(1,-1,0) + C₂(4, 3, 1) + c3(0, 1, 6) = (-6, 0, 5)