Vector Spaces: Direct Tensor Notation
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Problem D: Using direct tensor notation (ie., without resorting to component forms) prove that if
u, v, w are real numbers f V_3 and L, B are real numbers of R
a) (L u B v) = L B (u v)
b) (u (v + w)) = (u v) + (u w)
Hint: Take each expression and operate on a vector alpha is a real number of V_3 and proceed accordingly. This si often a useful strategy.
Problem E: Using direct tensor notation, prove
u A * B v = u * A(B v)
Problem F: A tensor T transforms every vector into its mirror image with respect to the plane whose normal is
n = sqrt(2)/2 (e_1 + e_2)
a) Find the components of T w.r.t. the basis {e_k}.
b) Use this tensor to find the mirror image of
a = e_1 + 2e_2
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