et A e C™×n; that is, A corresponds to a linear operator C" → C". efine the Hermitian transpose A* of A via the property that (Ах) у — х (А"у) for all x, у Є С" (Ac. ecall that here we need to use the inner product on C", which is the complex dot roduct defined by x · y - x "y = ΣΗ1 0.1). := X ) Prove that the property Ac uniquely defines the matrix A*, by showing that the natrix entries are given by (A*)ij = aji. Hint: use x = e¿, y = ej, where ej is the jth unit basis vector with (e;)k = 8jk-] i) Going the other way: define AH A")ij = aji), and show that AT (that is, let A" be the matrix with entries %3D

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

Determine if the vector product <A,B> = a0b0 defines a valid inner product on P2(R) for any A=a0+a1x+a2x^2 and B=b0+b1X+b2x2. Explain your answer clearly.

1. Let V₁, V2,...,Vn be n vectors in a vector space V. Explain what it means to say that these vectors span V.

Please help me.

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.

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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)

Each quadratic tensor can uniquely decompose into symmetric and asymmetric part.